REESE  LIBRARY 

OF    THE 

UNIVERSITY    OF   CALIFORNIA 

Received 
Accessions  No.       -^  <#.  Shelf  No. 


BARTLETT'S 
SPHERICAL  ASTRONOMY. 


ELEMENTS 


NATURAL   PHILOSOPHY 


BY 


W.  H.  C.  BARTLETT,  LL.D., 

PROFESSOR  OF  NATURAL  AND  EXPERIMENTAL  PHILOSOPHY  IN  THE  UNITED 
MILITARY  ACADEMY  AT  WEST  POINT, 

AUTHOR  OF 
"ELEMENTS   OF  MECHANICS,"    "ACOUSTICS,"    "OPTICS,"   AND 

"ANALYTICAL  MECHANICS." 


IV.— SPHERICAL  ASTRONOMY. 


FIFTH    EDITION,    REVISED    AND    CORRECTED. 

NEW   YOKE: 
A.  S.  BARNES  &  CO.,  Ill  &  113  WILLIAM  ST.,  COR.  OF  JOHN 

•OLD  BY  BOOKSELLERS  GENERALLY,  THROUGHOUT  THE  UNITED  STATES. 


Valuable  forte  liy  Leatii  Antlers 

TN  THE 

HIGHER   MATHEMATICS, 

W.  H.  C.  BARTLETT,  LL.D., 

'Prof,  of  Nat.  £  Exp.   Philog.  in  the  U.  ^.  Military  Academy,   West  Point* 
BARTLETT'S    SYNTHETIC    MECHANICS. 

Elements  of  Mechanics,  embracing  Mathematical  formulae  for  observing  and  calculating 
the  action  of  Forces  upon  Bodies — the  source  of  all  physical  phenomena. 

BARTLETT'S    ANALYTICAL    MECHANICS. 

For  more  advanced  students  than  the  preceding,  the  subjects  being  discussed  Analytically 
by  the  aid  of  Calculus. 

BARTLETT'S    ACOUSTICS    ANT)    OPTICS. 

Treating  Sound  and  Light  as  disturbances  of  the  normal  Equilibrium  of  an  analogous  char- 
acter, and  to  be  considered  under  the  same  general  laws. 

BARTLETT'S    ASTRONOMY*. 

Spherical  Astronomy  in  its  relations  to  Celestial  Mechanics,  with  full  applications  to  the 
current  wants  of  Navigation,  Geography,  and  Chronology. 

A.  E.  CHURCH,  LL.D., 

Prof.  Mathematics  in  the   United  States  Military  Academy,  West  Point. 

CHURCH'S    ANALYTICAL    G-EOMETRY. 

Elements  of  Analytical  Geometry,  preserving  the  true  spirit  of  Analysis,  and  rendering  11. e 
whole  subject  attractive  and  easily  acquired. 

CHURCH'S    CALCULUS. 

Elements  of  the  Differential  and  Integral  Calculus,  with  the  Calculus  of  Variations. 

CHURCH'S    DESCRIPTIVE    GEOMETRY. 

Elements  of  Descriptive  Geometry,  with  its  applications  to  Spherical  Projections,  Shades 
and  Shadows,  Perspective  and  Isometric  Projections.    2  vols. ;  Text  and  Plates  respectively. 


EDWARD   M.  COURTENAY,  LL.D., 

Late  Prof.  Mathematics  in  the  University  of  Virginia. 

COURTENAY'S    CALCULUS. 

A  treatise  on  the  Differential  and  Integral  Calculus,  and  on  the  Calculus  of  Variations. 


CHAS.  W.  HACKLEY,  S.T.  D., 

Late  Prof,  of  Mathematics  and  Astronomy  in  Columbia  College. 
HACKLE  Y'S    TRIGONOMETRY. 

A  treatise  on  Trigonometry,  Plane  and  Spherical,  with  its  application  to  Navigation  and 
Surveying,  Nautical  and  Practical  Astronomy  and  Geodesy,  with  Logarithmic,  Trigonomet- 
rical, and  Nautical  Tables. »«««•••« 

DAVIES   &   PECK, 

"Department  of  Mathematics,  Columbia  College. 
MATHEMATICAL    DICTIONARY 

And  Cyclopedia  of  Mathematical  Science,  comprising  Definitions  of  all  the  terms  employed 
in  Mathematics— an  analysis  of  each  branch,  and  of  the  whole  as  forming  a  single  science. 

C  H  ARLES   DAVIES,  L  L.  D., 

Late  of  the   United  States  Military  Academy  and  of  Columbia  College. 

A.    COMPLETE    COURSE    IN    MATHEMATICS. 

See  A.  S.  BABNES  &  Co.'s  Descriptive  Catalogue. 

Entered,  according  to  Act  of  Congress,  In  th.e  year  1859,  by 

W.  H.  C.  BARTLETT, 
In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District  of  New  York 

B'S  A'MY. 


PREFACE. 


THE  work  here  offered  to  the  public  was  undertaken  by  ita 
author  to  supply  a  want  long  felt  in  his  own  department  oi 
instruction  in  the  Military  Academy  at  West  Point.  Its  aim 
is  to  present  a  concise  course  of  Spherical  Astronomy  in  ita 
relationship  to  Celestial  Mechanics,  of  which  it  is  the  offspring. 
The  solar  and  stellar  systems  are,  therefore,  assumed  and  de- 
scribed as  necessary  facts,  arising  from  the  detached  condition 
of  the  bodies  which  compose  them  and  the  laws  of  universal 
gravitation.  The  consequences  from  these  systems,  to  a  spec- 
tator on  the  earth,  are  then  deduced,  and  their  entire  coinci- 
dence with  the  celestial  phenomena,  as  they  arise  spontane- 
ously, is  relied  upon  as  full  and  sufficient  justification  for  the 
assumption,  and  as  proof  that  the  systems  are  true.  This 
forms  the  first  part  of  the  subject.  A  general  account  of  the 
methods  by  which  the  future  condition  and  aspects  of  the 
heavens  are  predicted  follows,  and  the  more  important  appli- 
cations to  the  current  wants  of  Navigation,  Geography,  and 
Chronology,  conclude  the  volume. 


iv  PREFACE. 

In  the  description  and  discussions  of  instruments,  those  only 
have  been  selected  which  are  best  suited  to  convey  a  full  view 
of  the  whole  theory  and  practice  of  Astronomical  Measure- 
ments. 

The  author  would  acknowledge  his  obligation  to  Sir  John 
Ilerschel,  Professor  Challis,  Mr.  Maddy,  Mr.  Francis  Bailey 
Mr.  De  Morgan,  Mr.  Woolhouse,  M.  Francceur,  M.  De  Launay 
and  M.  Briot,  whose  works  have  been  constantly  before  him. 


CONTENTS. 


•MB 

Introductory  Remarks. 1 

Solar  System S 

Motion < 

Parallactic  Motion „ * 

Celestial  Sphere 7 

Shape  of  the  Earth '. « 

Diurnal  Motion 9 

Definitions , It 

Instruments 38 

Proportions  of  Land  and  Water — Atmosphere 13 

Infraction 14 

Parallelism  of  the  Earth's  Axis,  and  Uniformity  of  the  Earth's  Diurnal  Motion....  18 

Upper  and  Lower  Diurnal  Arcs — Circumpolar  Bodies 18 

Terrestrial  Latitude  and  Longitude .20 

Figure  and  Dimensions  of  the  Earth 21 

Geocentric  Parallax -. £4 

Augmented  Horizontal  Diameters 28 

Distances  and  Dimensions  of  the  Heavenly  Bodies 29 

Kcliptic  . . . . 30 

Precession  and  Nutation 87 

Sidereal  Time 40 

Earth's  Orbit 41 

Mean  Solar  Time 4« 

Aberration 50 

Heliocentric  Parallax 53 

The  Seasons 58 

Trade-Winds 59 

Terrestrial  Magnetism 62 

Tides 66 

Twilight 72 

The  Sun 78 

Planets 84 

Elements  of  the  Planets 85 

Dimensions  and  Distances  of  Planets  . .  . .  »0 


VI  CONTENTS. 

PMft 

Interior  Planets — Superior  Planets 90 

Synodic  Revolutions — Geocentric  Motions 91 

Direct  and  Retrograde  Motions— Stations 92 

Phases  of  the  Planets 93 

Transits — Occupations 95 

Masses  and  Densities  of  the  Planets 97 

Mercury 99 

Venus 1 00 

Mars— Planetoids 102 

Jupiter — Saturn 104 

Uranus — Neptune , 108 

Secondary  Bodies 109 

The  Moon— Lunar  Orbit 11G 

Disturbing  Forces 113 

Librations 115 

Lunar  Periods 116 

Lunar  Phases 117 

Eclipses  of  the  Sun  and  Moon 118 

Moon's  Relative  Geocentric  Orbit 123 

Ecliptic  Limits ; . . .   124 

Number  of  Eclipses 125 

The  Saros 126 

Physical  Constitution  oi  the  Moon '.27 

Satellites  of  Jupiter 1^4 

Progressive  Motion  of  Light oi 

Satellites  of  Saturn 134 

Satellites  of  Uranus 136 

Satellites  of  Neptune 137 

Crmets 137 

Elements  of  the  Orbits  of  the  Permanent  Comets 140 

Stars 144 

Elements  of  Stellar  Orbits •. 156 

Proper  Motion  of  the  Stars  and  of  the  Sun 158 

Nebulae , 15'J 

Kodiacal  Light 162 

Aerolites— -Meteors 163 

Ephemerides . .  165 

Catalogue  of  Stars 170 

Applications 174 

Time  of  Conjunction  and  of  Oppcsition , 174 

Angle  of  Position 175 

Projection  of  a  Solar  Ecrpse 175 

Projection  of  a  Lunar  Eclipse v 183 

Time  of  Day 184 

Azimuths 1S3 

Meridian  Passages 191 

Reduction  to  the  Meridian 193 

Terrestrial  Latitude 195 

Terrestrial  Longitude , 208 

Calendar..  ..  229 


CONTEXTS 


AI-PEADIX  I. — Elements  of  the  Principal  P'.anets 23f> 

APPENDIX  II. — Astronomical  Instrumc  :itP 237 

Clock  and  Chronometer 237 

Vernier 243 

Micrometer , 245 

Level 250 

Reading  Microscopes 252 

Transit 255 

Collimating  Telescope 266 

Vertical  Collimator 267 

Collimating  Eye-piece 268 

Mural  Circle 269 

Altitude  and  Azimuth  Instrument 276 

Equatorial 282 

Heliometer 293 

Sextant 294 

Artificial  Horizon * 298 

Principle  of  Repetition 300 

Reflecting  Circle 301 

APPENDIX  III. — Atmospheric  Refraction 305 

APPENDIX  IV. — Shape  and  Dimensions  of  the  Earth 31C. 

APPENDIX  V.— The  Earth's  Orbit 313 

APPENDIX  VI. — Planets'  Elements 316 

APPENDIX  VII. — Planets'  Elements 317 

APPENDIX  VIII.— Planets'  Elements 318 

APPENDIX  IX. — Planets'  Elements 31? 

APPENDIX  X. — Geocentric  Motion 331 

APPENDIX  XL — Mr.  Woolhouse  on  Eclipses,  &c 332 

APPENDIX  XIL— Equation  of  Equal  Altitudes 424 

APPENDIX  XIII. — Correction  for  Difference  of  Refraction 42* 


TABLES. 

TABLE  I.  —  Mr.  Ivory's  Mean  Refractions,  with  the  Logarithms  and  their  Differ- 

ences annexed  ..............................................  427 

TABLE  II.  —  Mr.  Ivory's  Refractions  continued:  showing  the  Logarithms  of  the 
corrections,  on  account  of  the  state  of  the  Thermometer  and 
Barometer  ..................................................  430 

TABLE  III.  —  Mr.  Ivory's  Refractions  continued  :  showing  the  further  quantities 
by  which  the  Refraction  at  low  altitudes  is  to  be  corrected,  on 
account  of  the  state  of  the  Thermometer  and  Barometer  .......  431 

TAISI.K   1  V.—  For  the  Equation  of  Equal  Altitudes  of  the  Sun  ....................   432 

V.  —  For    the    Reduction    to    the    Meridian:     showing    the    val.ie    of 


_ 

sill   V 

TAHLE  VI.  —  For  the  second  part  of  the  Reduction  to  the  Meridian  :  showing  the 

'2  sin*  t  P 

value  of  B  =  —         7—  ................  ......................  450 

sin  I" 


The  Greek  Alphabet  is  here  inserted  to  aid  those  who  are  not  already 
amiliar  with  it  in  reading  the  parts  of  the  text  in  which  its  letter  occur : 


Letters. 

Names. 

Letters. 

Name*. 

A  a 

Alpha 

N  v 

Nu 

B  j8e 

Beta 

S  I 

Xi 

r  yr 

CJamraa 

0  o 

Omicroo 

^  «5 

Delta 

n  or  « 

Pi 

K  s 

Epsilon 

V          r  P£ 

Rho 

7  ?£ 

V  Zeta 

2   *£ 

Sigma 

H    9) 

Eta 

T    T? 

Tau 

0  S£ 

Theta 

*jf  u 

Upsilon 

I    i 

Iota 

*    0 

Phi 

K  * 

Kappa 

X  x 

Chi 

A  X 

Lambda 

Y  4, 

Fsi 

M  M 

Mu 

r*  <-» 

Omega 

CONTENTIONAl    SlGNS    USED    IN    AsTRONOBlV. 

L,  for  mean  longitude,. 
M,  —  mean  anomaly,. 

Vr  —  true  anotro»}y, 

ft,  —  mean  daily  sidereal  motion^ 

r,  —  radius  vector, 

p,  —  angle  of  eccentpicityr 

<,  —  longitude  of  perihelionr 

a,  —  right  ascension, 

£,  —  declination, 

A,  —  logarithm  of  distance  frona  the  eartliy 

/,  —  heliocentric  longitude, 

6,  —  heliocentric  latitude, 

X,  —  geocentric  longitude, 

ft  —  geocentric  latitude, 

8,  —  longitude  of  ascending  nodef 

i,  —  inclination  of  orbit  to  the  ecliptic,  % 

angular  distance  from  perihelion  to  node, 
distance  from  noder  or  argument  for  latitude. 


ASTRONOMY, 

^ 

UNIVEBSl 


ASTRONOMY 

§  1.  THE  science  which  treats  of  the  heavenly  bodies  is  called  Astron- 
omy. It  is  divided  into  Physical  and  Spherical  Astronomy. 

§  2.  Physical  Astronomy  is  a  system  of  Mechanics,  in  which  the  forces 
are  universal  gravitation  and  inertia,  and  the  objects  the  gigantic  masses 
that  move  through  indefinite  space.  It  treats  of  the  physical  conditions 
of  the  heavenly  bodies,  their  mutual  actions  on  each  other,  and  explains 
tiie  causes  of  the  celestial  phenomena. 

§  3.  Spherical  Astronomy  is  mainly  concerned  with  the  appearances, 
magnitudes,  motions,  arrangements,  and  distances  of  the  heavenly  bodies ; 
and  seeks  to  apply  the  deductions  from  these  to  the  practical  wants  of 
society.  It  is  a  science  of  observation,  and  its  principal  means  of  investi- 
gation are  Optical  and  Mathematical  Instruments.  This  branch  of  As- 
tronomy will  form  the  subject  of  the  present  volume. 

§  4.  No  subject  calls  more  strongly  upon  the  student  to  abandon  first 
impressions  than  Astronomy.  All  its  conclusions  are  in  striking  contra- 
diction to  those  of  superficial  observation,  and  to  what  appears,  .at  first 
view,  the  most  positive  evidence  of  the  senses. 

§  5.  Every  student  approaches  it  for  the  first  time  with  a  firm  belief 
that  he  lives  on  something  fixed,  and,  abating  the  inequalities  of  hill  and 
"alley,  that  this  something  is  a  flat  surface  of  indefinite  extent,  composed 
of  land  and  water ;  and  that  the  blue  firmament  which  he  sees  around 
and  above  him  in  the  distance  is  a  stationary  vault,  upon  the  surface  of 
which  appear  to  be  placed  all  objects  out  of  contact  with  the  ground. 

§  6.  The  Earth  on  which  he  stands  is  divested  by  Astronomy  of  its 
flattened  shape  and  of  its  character  of  fixidity,  and  is  shown  to  be  a 
globular  body  turning  swiftly  about  its  centre,  and  moving  onward  through 
space  with  great  rapidity.  It  teaches  him  that  his  vault  has  no  existence 


2  ASTRONOMY. 

in  fac.t,  and  is  but  a  mere  illusion  which  comes  from  looking  through  the 
indefinite  space,  extended  without  limit,  in  which  he  is  moving. 

§  7.  Were  the  Earth  reduced  to  a  mere  point,  and  a  spectator  placed 
upon  it,  he  would  see  around  him  at  one  view  all  the  bodies  which  make 
up  the  visible  universe  ;  and  in  the  absence  of  any  means  of  judging  of 
their  distances  from  him,  would  refer  them  in  the  direction  in  which  they 
were  seen  from  his  station,  to  the  concave  surface  of  an  imaginary  sphere, 
having  its  centre  at  his  eye  and  its  surface  at  some  vast  and  indefinite 
distance. 

SOLAR  SYSTEM. 

§  8.  A  little  observation  would  lead  him  to  conclude  that  by  far  the 
greater  number  of  these  bodies  appear  fixed  while  the  rest  seem  ever  on 
the  move,  continually  shifting  their  positions  with  respect  to  those  which 
appear  fixed,  and  to  each  other.  The  former  are  called  FIXED  STARS  : 
the  latter  compose  what  is  called  the  SOLAR  SYSTEM,  a  group  of  bodies 
from  which  the  fixed  stars  are  so  remote  as  to  produce  upon  it  no  appre- 
ciable influence. 

§  9.  All  bodies  attract  one  another  with  intensities  which  are  propor- 
tional to  the  quantity  of  the  attracting  masses  directly,  and  to  the  squares 
of  the  distances  inversely,  Analyt.  Mech.,  §  205. 

§  10.  Bodies  resist  by  their  inertia  all  change  in  their  actual  state  of. 
motion ;  this  resistance  is  exerted  simultaneously  with  the  change,  and  i« 
always  equal  in  intensity,  and  contrary  in  direction,  to  the  force  which 
produces  it. 

§  11.  The  bodies  of  the  solar  system  have  motions  that  carry  them  in 
directions  oblique  to  the  lines  along  which  their  mutual  attractions  are 
exerted.  The  attractive  forces  draw  them  aside  from  these  directions ; 
inertia  resists  by  an  equal  and  contrary  reaction ;  and  the  bodies  are 
forced  into  curvilinear  paths,  and  made  to  revolve  about  the  centre  of 
inertia  of  the  whole. 

§  12.  Thus,  the  antagonistic  forces  of  gravitation  and  of  inertia  are  the 
simple  but  efficient  causes  which  keep  the  bodies  of  the  solar  system  to- 
gether as  a  single  group,  and  impress  upon  it  a  character  of  stability  and 
perpetuity.  But  for  the  force  of  gravitation  the  bodies  would  separate 
more  and  more,  and  wander  through  endless  space  ;  and  but  for  the  force  ( f 
inertia,  that  of  gravitation  would  pile  them  together  in  one  confused  mass. 

§  13.  The  force  of  gravitation  increases  rapidly  with  a  diminution,  and 
decreases  as  rapidly  with  an  augmentation,  of  distance.  Those  bodies 
which  are  nearest  exert,  therefore,  the  greatest  influence  upon  one  another's 


SOLAR   SYSTEM.  3 

motions.  Bodies  composing  an  insulated  group  may  perform  their  evolu- 
tions among  each  other  undisturbed  by  the  action  of  those  without,  pro- 
vided the  distances  of  the  latter  be  very  great  in  comparison  to  those 
which  separate  the  individuals  of  the  group. 

§  14.  This  is  a  characteristic  of  the  solar  system.  Its  own  dimensions, 
vast  as  they  are  when  expressed  in  terms  of  any  linear  unit  with  which 
we  are  familiar,  are  utterly  insignificant  when  compared  with  its  distance 
from  the  fixed  stars.  Each  of  the  latter,  by  virtue  of  this  relatively  great 
distance,  acting  upon  all  the  bodies  of  the  system  equally  and  in  parallel 
directions,  the  effect  of  the  whole  can  only  be  to  move  the  group  collec- 
tively through  space. 

§  15.  The  same  thing  takes  place  upon  a  smaller  scale  within  the  solar 
system  itself.  Some  of  its  members  are  so  close  together,  and  at  the  same 
time  so  far  removed  from  the  others,  as  to  be  forced  to  revolve  about  one 
another,  while  the  combined  action  of  the  rest  carries  them  as  a  sub-group, 
so  to  speak,  about  the  centre  of  inertia  of  the  whole. 

§  16.  The  mass  of  the  sun  so  far  exceeds  the  sum  of  the  masses  of  all 
the  other  bodies  of  the  system,  as  to  throw  the  centre  of  ineuia  of  the 
whole  group  within  the  boundary  of  its  own  volume ;  and  although  the 
centre  of  the  sun  actually  revolves  about  this  point,  yet  its  motion  bo- 
tomes  so  small,  when  viewed  from  the  distance  of  the  earth,  that  it  is  in- 
sensible except  through  the  medium  of  the  most  refined  instruments.  All 
the  other  bodies  are,  therefore,  said  to  revolve  about  the  sun  as  a  centre, 
and  it  is  from  this  fact,  and  the  controlling  influence  which  this  latter 
body  exerts  over  the  motions  of  all  the  others,  that  the  system  takes  its 
name. 

§  1 7.  The  same  is  true  of  the  sub-groups ;  the  mass  of  one  of  the  bodies 
in  each  being  so  much  greater  than  the  sum  of  the  masses  of  the  rest  as 
to  cause  the  latter  to  revolve  approximately  about  its  centre,  while  this 
centre  revolves  about  the  sun. 

§  18.  The  path  a  body  describes  about  another  as  a  principal  source  of 
attraction,  is  called  an  orbit. 

§  19.  Those  bodies  which  describe  their  orbits  about  the  stn  aie  called 
primary,  and  those  which  describe  their  orbits  about  the  primaries  are 
called  secondary  bodies.  These  latter  are  also  called  Satellites. 

Of  the  primary  bodies  there  are  three  distinct  classes,  differing  from 
each  other  mainly  in  the  shape  of  their  orbits,  their  densities,  and  gen- 
eral aspects. 

§  20.  A  body  subjected  to  the  action  of  a  central  force,  whose  intensity 
varies  as  the  square  of  the  distance  inversely,  mu«t  describe  one  or  other  of 


4  ASTRONOMY 

fhe  conic  sections,  depending  upon  the  relation  Between  its  velocity  and 
the  intensity  of  the  central  force.  The  orbits  that  are  known  to  belong  to 
the  solar  system  are  ellipses. 

§  21.  Those  primaries  which  move  in  elliptical  orbits  of  small  eccentri- 
cities are  called  PLANETS.  Those  primaries  having  orbits  of  great  eccentri- 
cities are  called  COMETS.  Comets  are  also  distinguished  from  planets  in 
having  a  degree  of  density  so  low  as  to  give  some  the  appearance  more  of 
a  vapor  than  of  a  solid  body. 

§  22.  The  solar  system  consists  then  of  the  Sun,  Planets,  Comets,  and 
Satellites.  Setting  out  from  the  sun,  the  known  planets,  with  their  names, 
occur  in  the  following  order,  viz. :  Mercury,  Venus,  the  Earth,  Mars, 
then  a  class  called  the  Planetoids,  of  which  ninety-one  are  known  at  the 
present  time,  Jupiter,  Saturn,  Uranus,  and  Neptune.  See  Plate  I.,  Fig.  1. 

To  these  must  be  added  a  multitude  of  much  smaller  bodies  of  the 
nature  of  planetoids,  whose  existence  is  inferred  from  the  fact  that  some 
of  their  number  make  their  way  now  and  then  to  the  earth's  surface  under 
tfie  name  of  meteors. 

§  23.  It  would  be  utterly  impossible  to  give  within  the  narrow  limits  of 
an  octavo  page  a  graphical  representation  of  the  relative  dimensions  of  the 
solar  system  ;  and  to  aid  the  conceptions  of  the  student,  Sir  John  Herschel 
has  instituted  the  following  illustration,  viz.  :  On  any  well-levelled  field 
place  a  globe  two  feet  in  diameter ;  this  will  represent  the  sun ;  Mercury 
will  be  represented  by  a  grain  of  mustard-seed  on  the  circumference  of  a 
circle  164  feet  in  diameter  for  its  orbit ;  Venus  a  pea  on  the  circumference 
of  a  circle  284  feet  in  diameter ;  the  Earth  also  a  pea  on  the  circumference 
of  a  circle  430  feet  in  diameter;  Mars  a  rather  large  pin's  head  on  the 
circumference  of  a  circle  of  654  feet  diameter ;  the  Planetoids  grains  of 
sand  on  circular  orbits  varying  from  1000  to  1200  feet  in  diameter; 
Jupiter  a  moderate  sized  orange  on  a  circumference  nearly  half  a  mile 
in  diameter ;  Saturn  a  small  orange  on  the  circumference  of  a  circle  four- 
fifths  of  a  mile  in  diameter ;  Uranus  a  full  sized  cherry  on  the  circumfer- 
ence of  a  circle  more  than  a  mile  and  a  half  in  diameter ;  and  Neptune 
a  good  sized  plum  on  the  circumference  of  a  circle  about  two  miles  and  a 
half  in  diameter.  To  illustrate  the  relative  motions,  Mercury  must  describe 
a  portion  of  its  orbit  equal  in  length  to  its  own  diameter  in  41  seconds; 
Venus  in  4  minutes  and  14  seconds ;  the  Earth  in  7  minutes  ;  Mars  in  4 
minutes  and  48  seconds ;  Jupiter  in  2  hours  and  56  minutes ;  Saturn  in  3 
hours  and  13  minutes;  Uranus  in  2  hours  and  16  minutes,  and  Neptune 
in  3  hours  and  30  minutes.  Now  conceive  the  two  feet  globe  to  be  in- 
creased till  its  diameter  becomes  880,000  English  miles,  and  suppose  the 


Plate  I. 


TO  FttOATT    PAW    4. 


SOLAR   SYSTEM.  5 

other  bodies  and  their  distances  increased  in  the  same  proportion  ;  the  re- 
sult will  represent  the  dimensions  of  the  solar  system.  It  will  give  to 
the  earth  a  diameter  of  nearly  eight  thousand  miles,  a  distance  from  the  sun 
equal  to  95  millions  of  miles,  and  a  velocity  through  space,  around  the  sun, 
of  19  miles  a  second. 

The  orbits,  although  referred  to  as  circles,  are  in  fact  ellipses,  but  of  ec- 
centricities so  small  as  to  justify  the  substitution  for  the  mere  purposes  of 
the  illustration. 

§  24.  The  fixed  stars  are  self-luminous.  The  sun  is  regarded  as  one  of 
this  class  of  bodies,  and  by  its  greater  proximity  to  the  earth,  becomes  the 
principal  source  of  heat  and  light  to  its  inhabitants. 

§  25.  The  planets  and  satellites  are  opaque  non-luminous  bodies,  and 
are  visible  only  in  consequence  of  light  received  from  the  sun  and  reflected 
to  the  earth* 


SPHERICAL  ASTRONOMY. 


MOTION. 

§  26.  Motion  signifies  the  condition  of  a  body,  in  virtue  of  which  it  oc 
cupies  successively  different  places.  But  we  can  form -no  idea  of  place  ex- 
cept by  referring  it  to  other  places,  and  these  again,  to  be  known,  must  be 
referred  to  others,  and  so  without  limit ;  so  that  place  is,  in  its  very  nature, 
entirely  relative.  Motion  is,  from  its  definition,  therefore,  also  relative. 

§  27.  We  judge  of  the  rate  of  motion  by  the  greater  or  less  rapidity 
with  which  the  object  possessing  it  varies  its  distance  from  other  objects 
assumed  as  origins.  These  origins  may  themselves  be  in  motion,  but  if 
the  circumstances  of  the  spectator  be  such  as  to  deceive  him  into  the  belief 
that  they  are  at  rest,  he  will  attribute  all  change  of  distance  to  a  motion 
wholly  in  the  object  which  he  refers  to  them.  And  this  is  one  of  the  most 
fruitful  sources  of  the  many  erroneous  notions  with  which  students  gener- 
erally  commence  the  study  of  astronomy. 

§  28.  If  two  objects  be  in  motion,  and  they  alone  occupy  the  spectator's 
field  of  view,  the  effect  to  him  will  be  the  same  if  he  suppose  one  fixed,  and 
attribute  the  whole  of  its  motion  to  the  other  in  a  contrary  direction  ;  for 
this  will  not  alter  the  rate  by  which  they  approach  to  or  recede  from  one 
another. 

PARALLACTIC  MOTION  AND  PARALLAX. 

§  29.  The  real  motion  of  a  spectator  gives  rise  to  the  appearance  of 
motion  among  surrounding  objects  which  are  relatively  at  rest.  Objects 
in  front  of  him  seem  to  separate  from  one  another,  those  behind  appear  to 
approach  one  another,  and  those  directly  to  the  right  and  left  seem  to  move 
in  a  direction  parallel  to  his  own  motion. 

A  spectator,  for  example,  travelling  over  a  plain  studded  with  trees  or 
other  objects  will,  on  fixing  his  eyes  upon  a  single  object  without  with- 
drawing his  attention  from  the  general  landscape,  see  or  think  he  sees  the 


CELESTIAL    SPHERE.  y 

latter  in  rotary  motion  about  that  object  as  a  centre ;  all  objects  between 
it  and  himself  appearing  to  move  backward,  or  contrary  to  his  own  motion, 
and  all  beyond  it,  forward  or  in  the  direction  in  which  he  moves. 

This  apparent  change  in  the  relative  places  of  objects,  arising  from  a 
shifting  of  the  point  of  view  from  which  they  are  seen,  is  called  parallactic 
motion  ;  and  the  amount  of  angular  change  in  the  instance  of  any  partic- 
ular object  is  called  the  parallax  of  that  object. 

§  30.  Let  P  be  the  place  of  an  object,  C  and  S  the 
places  from  which  it  is  seen ;  and  let  its  place  be  referred 
to  some  point  Z',  on  the  prolongation  of  the  line  CS, 
which  joins  the  points  of  view.  The  angular  change  in  the 
place  of  P  as  seen  from  C  and  S  will  be 

Z'  SP-Z'  CP=SP  C=  the  parallax  of  P. 

That  is  to  say,  the  parallax  of  an  object  is  the  angle  sub- 
tended at  the  object  by  the  distance  between  the  stations 
from  which  it  is  seen. 

Make  CP=d;  <7S=p;  the  angle  Z' SP=Z;  the  angle  SPC=z 
Then  from  the  triangle  C  S  P,  we  have 


sin  2=  ^  .  sin  Z 
d 


Whence  the  parallax  increases  with  an  increase  of  the  spectator's  change 
of  place,  with  diminution  of  the  object's  distance,  and  also  with  the  approx 
imation  of  Z  to  90°. 

§  31.  All  other  things  being  equal,  the  parallax  will  be  less  as  the  ob- 
ject's distance  is  greater ;  and  when  the  parallax  is  zero  for  any  arbitrary 

value  of  Z,  the  factor  -  must  be  zero,  and  the  change  of  the  spectator's 
place  must  be  utterly  insignificant  in  comparison  with  the  object's  distance 


CELESTIAL  SPHERE. 

§  32.  Now,  when  the  heavens  are  examined  it  is  found  that  by  far  the 
greater  number  of  the  celestial  bodies  have  no  sensible  parallax,  while 
comparatively  a  few  have.  The  first  are  the  fixed  stars  ;  and  they  are  so 
called  from  the  fact  that  they  always  preserve  the  same  angular  distances 
from  any  assumed  point  and  from  each  other,  from  whatever  station  on  the 
earth  they  are  viewed.  The  second  are  bodies  of  the  solar  system. 

§  33.  The  fixed  stars  are,  therefore,  beyond  limits  at  which  objects  cease 


8 


SPHERICAL    ASTRONOMY 


to  be  sensibly  affected  by  parallax.  The  great  concave  of  the  heavens 
upon  which  the  fixed  stars  appear  to  be  situated,  is  called  the  celestial 
sphere.  Not  only,  therefore,  is  the  longest  rectilineal  dimension  of  the 
earth,  but  also  the  distance  between  the  points  of  its  orbit  about  the  sun 
most  remote  from  each  other — a  distance,  as  we  shall  see  in  the  sequel, 
equal  to  one  hundred  and  ninety  millions  of  miles — utterly  insignificant 
when  expressed  in  terms  of  the  radius  of  the  celestial  sphere  as  unity.  A 
sphere  large  enough  to  contain  the  entire  orbit  of  the  earth  is  a  mere  point 
in  comparison  with  the  vast  volume  embraced  by  the  celestial  sphere. 
The  centre  of  ike  earth  may,  therefore,  always  be  regarded  as  the  centre  of 
the  celestial  sphere. 


Fig.  2. 


SHAPE  OF  THE  EARTH. 

§  34.  The  earth,  being  the  station  from 
which  all  the  other  heavenly  bodies  are 
viewed,  is  the  first  to  claim  attention.  It 
has  been  repeatedly  circumnavigated  in  dif- 
erent  directions,  and  the  portions  of  its  sur- 
face visible  from  elevated  positions  in  the 
midst  of  extended  plains  or  at  sea,  always 
appear  as  circles  of  which  the  spectator 
seems  to  occupy  the  centre.  The  apparent 
diameters  of  these  circles,  measured  by  in- 
struments, are  smaller  in  proportion  as  the 

points  of  view  S  are  more  elevated.  The  earth  is,  therefore,  ylobular  ; 
for  to  such  figures  alone  belong  the  property  of  always  presenting  to  the 
view  a  circular  outline. 

§  35.  By  the  figure  of  the  earth  is  meant  its  general  shape  without 
regard  to  the  irregularities  of  surface  which  form  its  hills  and  valleys. 
These  are  relatively  insignificant  and  are  disregarded  in  speaking  of  the 
earth's  form.  They  are  less  in  proportion  to  the  entire  earth  than  the 
protuberances  and  indentations  on  the  surface  of  a  smooth  orange  are  te 
a  large  size  specimen  of  that  fruit.  The  earth  is  an  oblate  spheroid,  and 
the  operations  and  method  of  computations  by  which  its  precise  magni 
tude  and  proportions  are  found,  will  be  given  presently. 

The  shortest  diameter  of  the  earth  is  called  its  axis. 


DIURNAL    MOTION. 


9 


DIURNAL  MOTION. 

§  36.  The  boundary  of  the  visible  portion  of  the  earth's  surface,  sup 
posed  perfectly  smooth,  is  called  the  sensible  horizon.    The  sensible  horizon 

Fig.  3. 


Fig.  4. 


is  only  seen  at  sea,  or  on  extended  plains.     At  most  localities  on  land  it  is 
broken  by  hills,  valleys,  and  other  objects. 

§  37.  The  earth  conceals  from  us  that  portion  of  space  below  our  sen- 
sible horizon,  while  all  above  is  exposed  to  view.  It  rotates  upon  its  axis, 
and  the  period  required  to  perform  one  entire  revolution  is  called  a  day. 

§  38.  Every  spectator  is  carried  about 
the  earth's  axis  in  the  circumference  of 
a  circle,  and  while  the  extent  of  the 
visible  portion  of  space  remains  un- 
changed, different  regions  are  continu- 
ally passing  through  the  field  of  view. 
The  horizon  of  a  spectator  will  be  ever 
depressing  itself  below  those  bodies 
which  lie  in  the  region  of  space  towards 
which  he  is  carried  by  the  rotation,  and 
elevating  itself  above  those  in  the  oppo- 
site quarter ;  thus  successively  bringing  into  view  the  former  and  hiding 
the  latter. 

§  39.  The  spectator  being  unconscious  of  his  own  motion,  concludes, 
from  first  appearances,  that  his  horizon  is  at  rest,  and  attributes  these 
changes  to  an  actual  motion  in  the  objects  themselves.  Instead  of  his 
horizon  approaching  the  bodies,  he  judges  the  bodies  to  approach  his 
horizon ;  and  when  it  passes  and  hides  them,  he  regards  them  as  having 
sunk  below  it  or  set,  while  those  it  has  just  disclosed,  and  from  which  it 
is  receding,  he  considers  as  having  come  up  or  risen. 

§  40.  One  entire  revolution  about  the  axis  being  completed,  the  spec- 
tator returns  to  the  place  from  which  he  commenced  his  observations,  and 
he  begins  again  to  witness  the  same  succession  of  phenomena  and  in  the 
same  order.  All  the  heavenly  bodies  appear  to  occupy  the  same  places 
m  the  concave  sky  which  they  did  before. 

§  41.  Thus  the  rotation  of  the  earth  about  its  axis  produces  the  daily 


10  SPHERICAL    ASTRONOMY 

rising  and  setting  of  the  sun — the  alternation  of  day  and  night ;  also  the 
rising  and  setting  of  the  other  heavenly  bodies,  their  progress  through  the 
vault  of  the  heavens,  and  their  return  to  the  same  apparent  places  at  short 
*nd  definite  intervals. 

§  42.  The  apparent  motions  with  reference  to  the  horizon  by  which 
these  daily  recurring  phenomena  are  brought  about,  are  called  the  diurnal 
motions  of  the  heavenly  bodies.  The  real  motion  is  in  the  horizon,  the 
origin  of  reference  ;  it  is  only  apparent  in  the  bodies  themselves. 

DEFINITIONS. 

§  43.  The  axis  of  the  celestial  sphere  is  the  axis  of  the  earth  produced. 

§  44.  The  poles  of  the  earth  are  the  points  in  which  its  axis  pierces  its 
surface.  The  pole  nearest  to  Greenland  is  called  the  north,  the  other  the 
south  pole. 

§  45.  The  poles  of  the  heavens  are  the  points  in  which  its  axis  pierces 
the  celestial  sphere.  That  above  the  north  pole  of  the  earth  is  called  the 
nor th,  the  other  the  south  pole. 

§  46.  The  earth's  equator  is  the  intersection  of  the  earth's  surface  by  a 
plane  through  its  centre,  and  perpendicular  to  its  axis.  . 

§  47.  The  equinoctial  is  the  intersection  of  the  surface  of  the  celestial 
sphere  by  the  same  plane. 

§  48.  A  meridian  line  is  the  intersection  of  the  earth's  surface  by  a 
plane  through  its  axis  and  the  place  of  a  spectator. 

§  49.  The  celestial  meridian  is  the  intersection  of  the  surface  of  the 
celestial  sphere  by  the  same  plane.  This  is  often  called  simply  the  me- 
ridian of  the  place. 

§  50.  The  poles  of  the  celestial  meridian  are  called  the  East  and  West 
points ;  that  towards  which  the  spectator  is  moving  by  his  diurnal  motion 
being  the  East,  that  from  which  he  is  receding  the  West. 

§  51.  The  apparent  zenith  and  apparent  nadir  are  the  points  in  which 
a  plumb-line  produced  intersects  the  celestial  sphere :  that  over  head  being 
the  zenith. 

§  52.  The  rational  horizon  is  the  intersection  of  the  celestial  sphere  by 
a  plane  through  the  earth's  centre  and  perpendicular  to  the  line  of  the 
zenith  and  nadir.  The  plumb-line  being  always  normal  to  the  earth's  sur- 
face, the  plane  of  the  rational  horizon  is  parallel  to  the  plane  tangent  to 
the  earth's  surface  at  the  spectator's  place,  and  these-  planes  intersect  the 
celestial  sphere  sensibly  in  the  same  great  circle. 

§  53.  The  dip  of  the  horizon  is   the  angle  which  the  elements  of  a 


DEFINITIONS.  jj 

visual  cone,  whose  vertex  is  in  the  eye  of  the  spectator,  and  whose  surface 
is  tangent  to  that  of  the  earth  along  the  sensible  horizon,  make  with  the 
tangent  plane  to  the  earth  at  the  spectator's  place.  The  dip  is  greater  in 
proportion  as  the  spectator's  elevation  above  the  earth  is  greater.  When 
the  eye  is  in  the  earth's  surface,  the  dip  is  zero,  and  the  visual  cone  be- 
comes the  tangent  plane.  This  coincidence  will  always  be  supposed  to 
exist  unless  the  contrary  is  specially  noticed. 

§  54.  The  latitude  of  a  place  on  the  earth's  surface  is  the  arc  of  the 
celestial  meridian  from  the  equinoctial  to  the  zenith  of  the  place.  It  is 
always  measured  in  degrees,  minutes,  seconds,  and  thirds.  Latitude  is 
reckoned  north  or  south ;  that  reckoned  towards  the  north  pole  being 
called  north  latitude,  that  towards  the  south  pole,  south  latitude.  The 
greatest  latitude  a  place  can  have  is  90°,  this  being  the  latitude  of  the 
poles  of  the  earth. 

§  55.  Parallels  of  latitude  are  small  circles  on  the  earth's  surface  par- 
allel to  the  equator.  All  places  on  the  same  parallel  have  the  same 
latitude. 

§  56.  The  longitude  of  a  place  on  the  earth's  surface  is  the  arc  of  the 
equinoctial  intercepted  between  the  meridian  of  the  place  and  that  of 
some  otlfer  place  assumed  as  a  first  meridian.  It  is  called  East  or  West, 
according  as  it  is  reckoned  in  the  direction  from  the  first  meridian  towards 
its  east  or  west  point.  For  the  sake  of  uniformity,  it  will,  in  the  texty  al 
ways  be  reckoned  in  the  latter  direction.  The  English  estimate  longitude 
from  the  meridian  of  Greenwich,  the  French  from  that  of  Paris,  and  other 
nations  from  other  meridians.  In  the  United  States,  for  most  geographical 
purposes,  it  is  estimated  from  the  meridian  of  Washington. 

§  57.  A  vertical  circle  is  the  intersection  of  the  celestial  sphere  by  a 
plane  through  the  zenith  and  nadir. 

The  prime  vertical  is  the  vertical  circle  whose  plane  is  perpendicular  to 
that  of  the  meridian. 

§  58.  The  north  and  south  points  are  the  poles  of  the  prime  vertical ; 
that  below  the  north  pole  being  called  the  north  point. 

§  59.  The  Azimuth  of  a  body  is  the  angle  which  a  vertical  circle 
through  the  body's  centre  makes  with  the  meridian.  It  is  measured  on 
the  horizon,  and  from  the  south  towards  the  west,  or  from  the  north  to- 
wards the  west,  according  as  the  north  or  south  pole  is  elevated  above  th« 
horizon.  It  may  vary  from  0°  to  360°. 

§60.  The  zenith  distance  of  an  objec  is  the  angular  distance  from 
the  apparent  zenith  to  the  centre  of  the  object,  measured  on  a  vertical 
circle. 


12  SPHERICAL    ASTRONOMY. 

§  61.  The  altitude  of  an  object  is  the  angular  distance  from  the  horizon 
to  the  object's  centre,  measured  on  a  vertical  circle. 

The  azimuth  and  zenith  distance  are  a  species  of  polar  co-ordinates  for 
the  designating  an  object's  place  in  the  heavens.  By  making  the  azimuth 
vary  from  zero  to  360°,  and  the  zenith  distance  from  zero  to  90°,  every 
visible  point  of  celestial  space  may  be  defined  in  position. 

§  62.  A  declination  circle,  or  hour  circle,  is  the  intersection  of  a  plane 
through  the  axis  of  the  heavens  with  the  celestial  sphere. 

§  63.  The  declination  of  an  object  is  the  angular  distance  of  its  centre 
from  the  equinoctial,  measured  on  a  declination  circle.  The  declination 
may  be  north  or  south,  and  may  vary  from  0°  to  90°. 

§  64.  The  polar  distance  of  an  object  is  the  angular  distance  of  its 
centre  from  the  celestial  pole,  measured  on  a  declination  circle. 

§  65.  The  right  ascension  of  an  object  is  the  angle  which  a  declination 
circle  through  the  object's  centre  makes  with  a  declination  circle  through 
a  certain  point  on  the  equinoctial,  called  the  Vernal  Equinox.  This  angle 
is  measured  upon  the  equinoctial,  and  eastwardly  in  direction. 

§  66.  The  polar  distance  and  right  ascension  are  also  a  kind  of  polai 
co-ordinates  for  defining  the  places  of  celestial  objects  ;  for  this  purpose  it 
is  only  necessary  to  cause  the  right  ascension  to  vary  from  0°  to  360°,  and 
the  polar  distance  to  vary  from  0°  to  180°,  to  reach  every  point  in  the 
celestial  sphere. 

§  67.  The  hour  angle  of  an  object  is  the  angle  which  its  hour  circta 
makes  with  the  meridian  of  the  place.  It  is  estimated  from  the  meridian 
westwardly,  and  may  vary  from  0  to  360°.  The  hour  angle  may  be  em- 
ployed, instead  of  the  right  ascension,  with  the  polar  distance  to  define  an 
object's  place. 

To  illustrate,  let  the  plane  of 
the  paper  be  that  of  the  meridian  ;  the 
circle  HZ  ON  its  intersection  with  the 
celestial  sphere;  P P'  the  axis  of  the 
heavens ;  P  and  P'  the  north  and  south 
poles  respectively  ;  Z  and  N  the  zenith 
and  nadir^  respectively,  and  the  earth  a 
mere  point  at  (7;  then  will  the  circle 
QWQ'E,  of  which  P  and  P'  are  the 
poles,  be  the  equinoctial;  HWOE,  of 
which  Z  and  N  are  the  poles,  the  hori- 
zon ;  E  and  W,  the  poles  of  the  meridian,  will  be  the  east  and  west  points 
respectively;  the  arc  ZQ  will  be  the  latitude,  ZSA  a  vertical  circle, 


INSTRUMENTS.  13 

Z  S  the  zenith  distance  of  the  object  S,  A  S  its  altitude,  and  0  WA  its 
azimuth  ;  PS  will  be  its  polar  distance,  D  S  its  declination,  Z P S,  meas- 
ured by  Q  />,  its  hour  angle,  and  if  V  be  the  vernal  equinox,  V D  will  be 

its  right  ascension. 

INSTRUMENTS. 

§  68.  Most  of  the  data  with  which  the  practical  astronomer  labors, 
come  from  measurements  made  in  the  circles  just  referred  to,  by  means 
of  certain  astronomical  instruments.  These  instruments  are  described, 
and  their  theory,  adjustments,  and  uses  explained,  in  Appendix  II. 
The  student  should  study,  in  connection  with  short  daily  lessons  of  the 
text,  from  this  point,  the  Clock.  Chronometer,  Transit,  Mural  Circle 
and  Azimuth  and  Altitude  Instrument.  The  others  should  be  taken 
up  where  referred  to,  in  the  order  of  the  text. 

PROPORTIONS  OF  LAND  AND  WATER.— THE  ATMOSPHERE. 

§  69.  To  resume  the  consideration  of  the  earth.  About  three  fourths 
of  its  surface  are  covered  with  water,  and  the  greatest  depth  of  the  sea 
does  not  probably  exceed  the  greatest  elevation  of  the  continents. 

The  earth  is  surrounded  by  a  gaseous  envelope,  called  the  atmosphere, 
the  actual  thickness  of  which,  were  it-  reduced  to  a  uniform  density 
throughout,  equal  to  that  at  the  surface  of  the  sea,  would  be  about 
five  miles.  But  owing  to  the  law  which  regulates  the  pressure,  density, 
and  temperature  of  elastic  bodies,  it  is  much  greater  than  this.  The  dif- 
ferent strata,  being  relieved  from  the  weight  of  those  below  them,  become 
more  expanded  in  proportion  as  they  are  higher,  and  the  place  of  the  su 
perior  atmospheric  limit  must  result  from  an  equilibrium  between  the. 
weight  of  the  terminal  stratum  and  the  elastic  force  of  that  upon  which 
it  rests.  The  laws  just  referred  to  indicate  that  this  limit  cannot  be  much 
higher  than  80  miles. 

§  70.  The  atmosphere  is  not  perfectly  transparent.  The  sun  illumines 
its  particles ;  these  scatter  by  reflection  the  light  they  receive,  particularly 
the  blue,  in  all  directions,  and  produce  that  general  illumination  called 
daylight  and  gives  to  the  sky  its  bluish  aspect.  But  for  this  diffusive 
power  of  the  air,  no  object  could  be  visible  out  of  direct  sunshine ;  the 
shadow  of  every  passing  cloud  would  be  pitchy  darkness,  the  stars  would 
Lo  visible  all  day,  and  every  apartment  into  which  the  sun  did  not  throw 
his  direct  rays  would  .be  involved  in  total  obscurity.  In  ascending  to 


SPHERICAL    ASTRONOMY. 


the  summits  of  high  mountains,  the  diffused  light  becomes  less  and  less, 
the  sky  deepens  in  hue,  and  finally,  at  great  altitudes,  approaches  to  total 
blackness. 

§  71.  The  superior  illumination  of  the  atmosphere  produced  by  the 
solar  light  obliterates,  as  it  were  by  contrast,  the  light  from  almost  all 
the  other  heavenly  bodies,  and  few,  if  any,  of  the  latter  are  seen  when 
the  sun  is  up. 

REFRACTION. 

§  72.  Luminous  waves  which  enter  the  atmosphere  obliquely  are,  ac- 
cording to  the  laws  of  optics,  deviated  by  the  latter  from  their  course,  and 
made  to  exhibit  the  objects  from  which  they  proceed  in  positions  different 
from  those  they  actually  occupy,  and  thus  false  impressions  are  produced 
in  regard  to  true  places  of  the  heavenly  bodies. 

Take,  for  example,  a  spectator  fig.  86. 

on  the  earth  at  A ;  and  let  L  D  L 
represent  a  section  of  the  supe- 
rior limit  of  the  atmosphere,  and 
KA  A'  that  of  the  earth's  sur- 
face by  a  vertical  plane.  A  star 
at  S  would,  in  the  absence  of 
the  atmosphere,  appear  in  the 
direction  A  S ;  but  in  reality, 
when  the  portion  of  the  luminous 
wave  moving  on  this  line  reaches 
the  point  J9,  it  is  turned  down- 
ward, and  made  to  come  to  the 
earth  at  some  point  A',  pursuing 
a  course  such  as  to  bring  its  suc- 
cessive positions  normal  to  some 

curve,  as  DA',  whose  curvature  increases  towards  the  earth's  surface,  in 
consequence  of  the  increasing  density  of  the  atmosphere  in  that  direction. 
This  part  of  the  wave  cannot  therefore  go  to  the  spectator.  Not  so,  how- 
ever, with  a  portion  of  the  same  general  wave  incident  at  some  point  as 
D\  nearer  to  the  zenith  ;  this,  after  pursuing  a  path  D'A  similar  to  DA', 
will  reach  the  spectator  at  A,  and  cause  the  body  from  which  it  originally 
proceeded  to  appear  in  the  direction  A  S',  tangent  to  the  curve  at  the 
point  A,  the  effect  being  the  same  as  though  the  body  had  shifted  its 
place  towards  the  zenith  by  the  angular  distance  S  A  S'. 

§     73.  The  air's  refraction,  therefore,  diminishes  apparently  the  zenith 


REFRACTION.  13 

distances  of  all  bodies,  and  increases  their  altitudes.  Any  body  actually  in 
the  horizon  will  appear  above  it,  and  any  body  apparently  in  the  horizon 
must  be  below  it. 

§  74.  It  is  also  obvious  that  refraction  can  only  take  place  in  the  ver- 
tical plane  through  the  body,  since  this  plane  is  always  normal  to  the 
surfaces  of  the  atmospheric  strata,  and  divides  them  symmetrically.  Re- 
fraction will  not,  therefore,  in  general,  affect  the  azimuth  of  a  body. 

§     75.  This  apparent  angular  displacement  of  a  body  from  its  true 
place,  caused  by  the  action  of  the  atmosphere  upon  its  luminous  waves,  is 
called  refraction  j  and  various  formulas  have  been  constructed  to  compute 
its  exact  amount.     One  of  the  best  of  these  is  by  Littrow,  which  has  the 
merit  of  depending  upon  no  special  hypothesis  in  regard  to  the  constitu- 
tion of  the  atmosphere,  being  constructed  upon  the  most  general  prinoi 
pies,  and  from  known  and  well-ascertained  data. 
§     76.  Make, 

Z  =  Zf  A  S'  =  observed  zenith  distance ; 
r  =  S  A  S1  =  corresponding  refraction  ; 

h  =  height  of  mercurial  column,  which  the  atmosphere  supports  ; 
t  =  temperature  of  the  air  and  of  the  mercury  ; 
a  —  coefficient  of  atmospheric  expansion  for  each  degree  of  Fahr. ; 
(3  =  coefficient  of  expansion  for  mercury,  same  thermometric  scale 

Then,  Appendix  No.  III., 

/  9.4-si'n2^V 

(2) 


'30'l  +  (*-50)a  ' 

or,  omitting  the  last  term  in  the  parenthesis  as  being  insignificant  for  or 
dinary  zenith  distances, 


r  =  57".82.—  ,±-  -^.  tan  Z  .(I  -0.0012517  sec'Z)  .  .    (3) 

30     1  +  (t  —  50ja 

When  h  =  30,  and  t  =  50,  equation  (  3  )  becomes 

rin  =  57".82  tan  Z  (1  -  0.0012517  sec2  Z)  =  A    .     .       (4) 

and  the  results  given  by  this  formula  for  different  values  for  Z  are  called 
mean  refractions ;  and  for  any  other  state  of  the  thermometer  and  barometer. 

h    1  +  (50  —  *)/3 
r  =  A'W'l  +  (t-5V)*> 
and  taking  logarithms, 

,   .        A  1  +  (50-0#  ,K\ 

logr=::log^  +  log-  +  logr^-—    .       .       .          (5) 


16  SPHERICAL   ASTRONOMY. 

Causing  Z  to  vary  from  0°  to  90°,  h  from  28  to  31  inches,  and  t  from 
80°  to  20°,  the  logarithms  above  may  be  computed  arid  tabulated  for 
future  use,  under  the  heads  Z,  t,  and  b. 

§  77.  Causing  Z  to  vary  from  0°  to  90°,  in  equation  (4),  we  may 
construct  Table  I.; causing  t  to  vary  from  80°  to  20°,  and  h  to  vary  from 
31  to  28,  in  the  last  two  terms  of  equation  (5),  we  may  construct  Table 
II.  Returning  to  equation  (2),  resuming  the  quantity  omitted  to  obtain 
equation  (3),  computing  their  values  for  zenith  distances,  varying  from 
75°  to  90°,  on  the  supposition  that  A=30  and  £=50,  an  additional  table 
may  be  computed  to  correct  the  refractions  in  low  altitudes.  Tables  L, 
II..  and  III.  are  due  to  Mr.  Ivory. 

§  78.  For  zenith  distances  exceeding  80°,  refraction  becomes  very 
uncertain  ;  it  then  no  longer  depends  solely  upon  the  state  of  the  atmo- 
sphere, which  is  indicated  by  the  barometer  and  thermometer,  being  fre- 
quently found  to  vary  at  the  same  station  some  3  to  4  minutes  for  the 
same  indications  of  these  instruments. 

Example. — The  zenith  distance  of  an  object  is  observed  to  be  71°  26'  00", 
the  barometer  standing  at  29.76  in.,  and  the  thermometer  at  43°  Fahr  : 
required  the  refraction. 

Table  I.  Mean  refraction,  log.  2.23609 

Table  II.         Barometer  29.76  "     9.99651 

Table  II.         Thermometer  43°  "     0.00668 

Hefraction 2'  53".49  .  .   2.23928 

Observed  zenith  distance  .  71°  26'  00".00 


Zenith  dist.  cleared  from  refraction  71°  28'  53 ".49 


The  refraction  must  always  be  added  to  the  observed  zenith  distance,  or 
subtracted  from  the  observed  altitude,  to  clear  an  observation  from  re- 
fraction. 

PARALLELISM  OF   THE    EARTH'S    AXIS,  AND   UNIFORMITY  OF  THE 
EARTH'S  DIURNAL  MOTION. 

§  79.  Wherever  upon  the  earth's  surface  the  altitudes  and  instru- 
mental azimuths  of  a  star  are  taken  in  the  various  points  of  its  diurnal 
course,  and  the  instrument  is  turned  in  azimuth,  so  as  to  read  the  half  sum 
of  two  azimuths,  corresponding  to  any  two  equal  altitudes,  the  vertical 
plane  through  the  line  of  collimation  is  found  to  divide  the  path  symmet- 


PARALLELISM    OF   THE    EARTH'S    AXIS.  17 

rically ;  and  this  plane  of  symmetry  for  any  one  star  will,  at  the  same 
place  of  observation,  also  be  a  plane  of  symmetry  for  all  the  stars.  In 
other  words,  the  diurnal  paths  of  the  stars  may  be  divided  symmetrically 
by  any  number  of  planes  inclined  to  one  another  through  the  earth's 
centre — a  condition  which  can  only  be  fulfilled  for  paths  upon  the  celes- 
tial sphere,  when  these  paths  are  circles,  of  which  the  poles  coincide,,  and; 
the  planes  qf  symmetry  pass  through  them. 

The  diurnal  motions  of  the  stars  are  only  apparent,  and  arise  from  an, 
actual  motion  of  the  spectator  about  the  earth's  axis.  This  latter  line 
preserves,  therefore,  its  direction  unchanged,  and,  in  the  motion  of  the 
earth  around  the  sun,  describes  a  cylindrical  surface,  of  which  the  elements 
have  their  vanishing  point  in  the  poles  of  the  celestial  sphere.  These 
poles  are  therefore  the  geometric  poles  of  the  diurnal  paths  of  the  stars, 
and  the  planes  of  symmetry  are  the  meridian  planes  of  the  places  of 
<  >bservation. 

§  80.  Again,  the  interval  of  time  during  which  a  star  is  moving  be 
tween  any  two  given  altitudes  on  one  side  of  the  plane  of  symmetry,  is 
exactly  equal  to  that  during  which  it  is  moving  between  the  equal  alti- 
tudes on  the  opposite  side,  which  can  only  be  true,  for  all  positions  of  the 
observer,  when  the  star's  apparent,  or  the  earths  real  motion  about  its 
axis,  is  uniform. 

§  81.  The  period  of  one  revolution  of  the  earth  about  its  axis  is  called 
H  day;  the  day  is  divided  into  24  equal  parts  called  hours;  the  hours 
into  60  equal  parts  called  minutes  ;  the  minutes  into  60  equal  parts  called 
seconds,  and  the  seconds  into  60  equal  parts  called  thirds. 

§  82.  The  earth  rotates  therefore  at  the  rate  of  360^-24  =  15°  an 
hour ;  15'  of  space  in  1  minute  of  time ;  15"  of  space  in  1  second  of  tiro^, 
or  15'"  of  space  in  1  third  of  time. 

§  83.  Distances  on  the  equinoctial  may  therefore  be  expressed  in 
time  or  space  at  pleasure,  the  former  being  convertible  into  the  latter  by 
multiplying  by  15,  or  the  latter  into  the  former  by  dividing  by  15. 

§  84.  To  distinguish  hours,  minutes,  and  seconds  in  time,  from  degrees, 
minutes,  and  seconds  in  arc,  the  formerare  usuall)  designated  by  the  nota 
tion  h,  m,  s,  and  the  latter  by  °,  ',  "  ;  thus  an  arc  upon  the  equinoctial 
may  be  written  357°  39'  38",  or  23h  50m  388.5. 

§  85.  To  find  the  instrumental  azimuth  of  the  meridian  of  a  place. — 
Bring  the  line  of  collimation  of  an  altitude  and  azimuth  instrument,  prop- 
erly levelled,  upon  a  star  in  the  east  or  west,  clamp  the  vertical  circle,  and 
read  the  instrumental  azimuth ;  then  by  an  azimuthal  motion  bring  the  line 
of  collimation  upon  the  star  when  in  the  west  or  east,  and  again  read  the 

2 


18 


SPHERICAL   ASTRONOMY. 


azimuth  :  the  half  sum  of  the  two  will  be  the  instrumental  azimuth  sought. 
To  bring  the  line  of  collimation  into  the  meridian,  turn  the  instrument  till 
it  reads  this  half  sum. 


UPPER  AND  LOWER  DIURNAL  ARCS.— CIRCUMPOLAR  BODIES. 


Fig.  87. 


§  86.  The  diurnal  paths  of  the  heavenly  bodies  which  are  cut  by  the 
horizon  are,  in  general,  divided  by  the 
latter  unequally.  The  portions  of  these 
paths  above  the  horizon  are  called  the 
upper,  and  those  below  the  lower  diurnal 
arcs. 

§  87.  To  find,  for  any  spectator,  thb 
relation  which  these  arcs  bear  to  one  an- 
other, let  PQP'Q'  be  the  meridian,  P 
the  elevated  pole,  Q  Q'  the  equinoctial, 
Z  the  zenith,  H  W  H'  the  horizon, 
S'SS"S"r  the  diurnal  path  of  any 
body,  the  earth  being  a  mere  point  at  E ; 
then  will  Sr  S  S"  be  the  upper,  and  S"  S"f  S1  the  lower  diurnal  arc. 

Make 

• 

/  =  Q  Z,  latitude  of  the  spectator, 
p  =  P  S",  polar  distance  of  the  body, 
P  =  ZP  S",  the  hour  angle  of  the  body  when  in  the  horizon, 

2  =  Z  S",  zenith  distance  of  the  body  in  horizon. 

Then  in  the  triangle  ZP  S",  because  PZ  —  -90  —  J, 

cos  z  =  cos  p  sin  I  +  sin  p  cos  I  cos  P     .     .     .     .       (6) 
but  2  =  90°,  whence 

0  ==  cos  p  sin  I  +  sin  p  cos  I  cos  P ; 
or 


cos  P  =  - 


tan  I 


If  I  =  0,  or  p  =  90°,  then  will 

cos  P  =  0,  and  P  —  90°  =  6h ; 

that  is,  if  the  spectator  be  upon  the  equator,  or  the  body  upon  the  equi- 
noctial, the  semi-upper  arc  will  be  six  hours,  and  the  body  will  be  a*  long 
above  as  below  the  horizon.- 


CIRCUMPOLAR    BODIES.  19 

If  p  <  J,  then  will 

cos  P  <  —  1 ; 

which  is  impossible,  and  the  place  of  the  body  can  never  satisfy  the  con- 
dition that  z  =  90°.  In  other  words,  when  the  jolar  distance  is  less 
than  the  latitude  of  the  spectator's  place,  the  body  can  never  sink  to  the 
horizon,  and  will  ever  remain  in  the  field  of  perpetual  apparition.  Such 
bodies,  as  well  as  their  diurnal  paths,  are  said  to  be  circumpolar. 
If  p  —  /,  then  will 

cosP— —  1;   P=180°=12h; 

that  is,  when  the  polar  distance  of  the  body  is  equal  to  the  latitude  of 
the  spectator's  place,  the  body  can  never  sink  below  the  horizon,  but  will 
iust  gra^ze  it  in  the  meridian. 
If  p  >  /,  and  p  <  90°, 

cos  P  <  0,  cos  P  >  —  1 ;   P  >  90°,  P  >  6h ; 

that  is,  all  bodies  between  the  elevated  pole  and  the  equinoctial,  will  be 
longer  above  than  below  the  horizon. 
If  p  >  /,  and  p  >  90°, 

cos  P  >  0,  cos  P  <  1,  P  <  90°,  P  <  6h ; 

that  is,  if  the  body  and  the  spectator  be  on  opposite  sides  of  the  plane  ol 
the  equinoctial,  the  semi-upper  arc  will  be  less  than  six  hours,  and  the 
body  will  be  a  shorter  time  above  than  below  the  horizon. 
If  p  =  180°  —  Z,  then  will  tan  p  =  —  tan  /,  and 

cos  P=  1,  P=00  =  0h; 

that  is,  when  the  body  is  at  a  distance  from  the  depressed  pole  equal  to 
the  latitude  of  the  place,  the  body  will  never  rise  above  the  horizon,  but 
just  graze  it  in  the  meridian. 

If  p  >  180°  —  I,  then  will  tan  p  >  -  tan  /,  and 

cos  P  >  1, 

which  is  impossible.     That  is  to  say,  if  the  body's  distance  from  the  de- 
pressed pole  be  less  than  the  spectator's  latitude,  the  body  can  never  ris 
to  the  horizon,  and  must  ever  remain  invisible. 

§  88.  The  act  of  a  body's  passing  the  meridian,  is  called  its  culmina 
tion.  A  body  has  its  greatest  or  least  altitude  at  the  instant  of  its  cul 
mination.  The  altitude  of  a  body  when  on  the  meridian  is  called  its 
meridian  altitude. 


SSO 


SPHERICAL    ASTRONOMY, 


TERRESTRIAL  LATITUDE  AND  LONGITUDE. 


§  89.  Latitude.— When  in  Eq.  (  7  )  the  angle 
then  will  p  =  I ;  but  in  this  case  p  is  the  polar  distance  of  the  point  of  the 
horizon  of  the  same  name  as  the  elevated  pole,  and  hence  the  latitude  of  the 
spectator  is  always  equal  to  the  altitude 
of  the  elevated  pole. 

§  90.  This  suggests  an  easy  and 
accurate  method  of  getting  from  obser- 
vation both  the  latitude  of  the  specta- 
tor's place  and  the  polar  distance  of  a 
star. 

Let  Z  be  the  zenith,  HIT  the  hori- 
zon, Q  Q'  the  equinoctial,  P  the  eleva- 
ted and  P'  the  depressed  pole,  and  S' S, 
the  diurnal  path  of  a  circumpolar  star. 
Make 

/  =  HP  =  Z  Q  ,  the  latitude, 
p  =  P  S'  =  P  S  ,  the  polar  distance  of  star, 
a'  —  H Sf  ,  the  greatest  observed  meridian  altitude  of  star, 

a{  =  HS  ,  the  least  observed  meridian  altitude  of  star, 

r'  and  rt  ,  the  refractions  corresponding  to  the  greatest  and 

least  meridian  altitudes  respectively. 

Then  from  the  figure  will 

^.i^a=i.i±i^E±aJ    ...    (8) 


P  = 


That  is  to  say,  the  latitude  of  the  observer's  place  is  equal  to  the  half  sum 
of  the  greatest  and  least  meridian  altitudes  of  a  circumpolar  star  ;  and  the 
polar  distance  of  the  star  is  equal  to  the  half  difference  of  its  greatest  and 
least  meridian  altitudes.  Other  methods  for  finding  the  latitude  will  be 
given  in  another  place. 

§  91.  Longitude. — The  uniform  motion*  of  the  earth  about  its  axis  ftir- 
nishes  the  means  of  finding  the  longitude  of  the  spectator's  place. 

Twenty-four  perfect  time-keepers,  with  dial-plates  graduated  to  24  hours, 
placed  upon  meridians  15°  apart,  and  so  regulated  as  to  mark  24h  at  the 
instant  any  one  fixed  star  or  other  point  of  the  heavens  culminates,  would, 


FIGURE   OF   THE   EARTH.  21 

§  82,  when  this  regulating  star  or  point  comes  to  any  one  of  these  me- 
ridians, simultaneously  mark  the  hours  indicated  by  the  natural  numbers 
from  one  to  twenty-four, inclusive;  that  15°  to  the  east  of  the  regulating 
point  marking  lh,  that  30°  to  the  east  marking  2h,  and  so  on  to  that  345° 
to  the  east,  or  1 5°  to  the  west,  marking  23h.  The  timepieces  to  the  east 
would  be  later  and  later,  those  to  the  west  earlier  and  earlier.  The  times 
indicated  on  these  several  timepieces  are  called  the  local  times  of  their  re- 
spective meridians. 

§  92.  If  now,  without  altering  its  hands  or  rate  of  motion,  a  traveller 
were  to  transport  the  time-keeper  of  any  one  of  these  meridians  to  that  on 
any  other,  and  note  the  difference  of  time  indicated  by  the  two,  this  differ- 
ence would  be  the  difference  of  longitude  of  the  two  meridians,  expressed 
in  time ;  and  multiplied  by  15  would  give  the  same  in  degrees. 

§  93.  If  one  of  these  meridians  be  the  first  meridian,  this  difference 
would  be  the  longitude  of  the  other.  But  if  neither  be  the  first  meridian, 
this  difference  applied  to  the  longitude  of  one,  supposed  known,  would  give 
the  longitude  of  the  other. 

§  94.  The  solution  of  the  problem  of  longitude  consists,  therefore,  in 
finding  the  difference  of  the  local  times  which  exist  simultaneously  on  the 
first  and  required  meridians.  The  various  modes  of  doing  this  will  be 
given  in  another  place. 


FIGURE  AND  DIMENSIONS  OF  THE  EARTH. 

§  95.  A  fluid  mass  rotating  about  an  axis,  and  of  which  the  particles 
attract  one  another  with  intensities  varying  inversely  as  the  square  of  their 
distances  apart,  will  assume  the  form  of  an  oblate  spheroid.  Its  axis  of 
rotation  will  be  both  the  shortest  and  a  principal  axis  of  figure.  Where 
the  angular  velocity  is  such  as  to  make  the  centrifugal  force  of  the  sur- 
face elements  small  in  comparison  with  their  weight,  due  to  the  attraction 
of  the  whole  mass,  the  figure  of  the  meridian  section  will,  (§  265,  Analyt. 
Mechanics,)  approach  that  of  an  ellipse  of  small  eccentricity. 

§  96.  The  centrifugal  force  of  a  body  at  the  equator  of  the  earth, 
where  it  is  greatest,  is  only  about  ?j-f  th  part  of  its  weight.  Observations 
upon  the  temperature  of  the  strata  composing  the  earth's  crust,  lead  to 
the  conclusion  that  at  no  great  depth  below  its  surface  its  materials  are  in 
a  fluid  state  from  excessive  heat ;  and  the  researches  of  geology  make  it 
more  than  probable  that  there  was  a  time  when  the  earth  was  without 
Solid  matter.  Its  present  irregularities  of  surface,  forming  mountains, 
hills,  valleys,  the  bed  of  the  ocean,  of  seas,  lakes  and  rivers,  are  due  to 


22 


SPHERICAL  ASTRONOMY. 


changes  subsequent  to  the  surface  induration  from  cooling,  and  as  the  ver- 
tical dimensions  of  these  are  insignificant  in  comparison  with  the  depth 
to  the  centre  of  the  entire  mass,  it  is  concluded  that  the  figure  of  the 
earth  is  one  of  fluid  equilibrium  due  to  its  rotary  motion. 

§     97.  Assuming  the  meridian  section  of  the  earth  to  be  an  ellipse,  its 
eccentricity  and  semi-axes  are  found,  Appendix  No.  IV.,  from  the  relations 


c  —  c' 


'  c  sin2 1'—  c'  sin8 


1-V 


(10) 


(H) 
(12) 


Fig. 


in  which 

e  =  the  eccentricity  of  the  meridian  ; 
A  =  semi-transverse  axis  =  equatorial  radius  of  the  earth  ; 
B  =  semi- conjugate  axis  =  polar  radius  of  the  earth  ; 
c  and  c'  =  the  linear  dimensions  of  the  arcs  of  the  meridian,  whose 

extremities  differ  in  latitude  by  1°  ; 

lm  and  l'm  —  latitudes  of  the  middle  joints  of  the  arcs  c  and  c'  respec- 
tively. 

The  quantities  lm,  l'm,  c,  cr  are  found  from  observation  and  measure- 
ment.    A  method  by  which  /  and  /'  may  be  found  is  explained  in  §  90. 

§  98.  To  find  c  and  e',  a  base  line  AB  is  carefully  measured  on  some 
extended  plain,  and  a  number  of  stations  (7,  D,  E,  F,  H,  <fee.,  are  se 
lected  in  a  northerly  or  southerly  direction,  and  so  that  C 
may  be  seen  from  A  and  J5,  D  from  B  and  (7,  E  from  C 
and  D,  and  so  on  to  the  end.  The  several  stations  being 
connected  by  right  lines,  a  network  of  triangles  is  formed  ; 
every  angle  in  each  triangle  is  carefully  measured,  and  the 
instrumental  azimuth  of  its  vertex,  and  that  of  the  meridian, 
as  viewed  from  the  other  two,  accurately  noted,  (§  85). 
The  angles  being  cleared  from  spherical  excess,  the  sides  of 
the  triangles  are  then  computed,  beginning  of  course  with 
the  triangle  of  which  the  measured  base  is  one  of  the  sides. 
The  difference  between  the  instrumental  azimuths  of  the 
.several  vertices  and  those  of  the  meridian,  gives  the  inclina- 
tion of  the  sides  to  the  meridian  line.  The  product  of  each 
side  into  the  cosine  of  its  inclination  gives  the  projection  of  this  side  on 
the  meridian,  and  the  sum  of  the  projections  of  any  one  of  the  series  of 
sides,  as  A  J5,  B  C,  CD,  1)  E,  EH,  and  H F,  connecting  the  most  north- 


FIGURE   OF   THE    EARTH  23 

erly  and  southerly  points,  will  give  the  linear  meridional  distance  L  L\ 
between  the  parallels  of  .atitude  through  the  same  points. 
Make 

a  =  the  sum  of  these  projections,  expressed  in  miles  ; 
ln  =  the  latitude  of  A,  supposed  the  most  northerly  ; 
Z,  =  the  latitude  of  F,  supposed  the  most  southerly  ; 
then, 

ln  —  lt:l°::a  :  c, 
whence 


and 


The  same  operations  being  repeated  in  a  different  locality  considerably 
further  north  or  south,  the  values  of  c'  and  l'm  are  found,  and  hence  from 
equations  (10),  (11),  and  (12),  the  dimensions  of  the  earth. 

From  the  arcs  known  as  the  Peruvian,  Indian,  French,  English,  Hano- 
verian, Danish,  Prussian,  Russian,  and  Swedish,  names  derived  from  the 
countries  in  which  the  arcs  were  mostly  measured,  Bessel  found, 

e2  =  0.0068468, 
2  A  =  7925,604  miles,  j 
2£  =  7899,114  miles,  V        ......     (13) 

Polar  compression  =      26,490  miles.  ) 

§  99.  By  the  ellipticity  of  the  earth  is  meant  the  difference  between 
its  equatorial  and  polar  radii,  expressed  in  terms  of  the  equatorial  radius 
as  unity.  Denoting  the  ellipticity  by  E,  we  have 


§  100.  The  length  of  a  degree  of  latitude,  denoted  by  /3,  in  any  lati- 
tude /,  is,  Appendix  No.  IV,  equation  (/),  given  by 


The  length  of  a  degree,  measured  perpendicularly  to  the  meridian,  de- 
noted by  |3,,  is,  Appendix  No.  IV.,  equation  (w),  given  by 


24:  SPHERICAL    ASTRONOMY. 

and  the  length  of  a  degree  of  longitude,  denoted  by  a,  measured  on  a  par- 
allel of  latitude  in  the  latitude  Z,  is,  App.  No.  IV.,  equatiou  (o),  given  by 


§  101.  The  close  agreement  between  the  results  of  these  formulas  and 
those  of  actual  measurement,  at  various  and  numerous  places  on  the  earth, 
justifies  in  the  fullest  manner  the  assumption  in  regard  to  its  ellipsoidal 
figure. 

The  equatorial  circumference  of  the  earth  is  24,899,  say,  for  convenience 
of  memory,  25,000  miles.  The  lengths  of  the  degrees  of  latitude  increase 
from  the  equator  to  the  poles.  In  the  latitude  of  50°  the  length  is  about 
70  statute  miles,  and  contains  nearly  as  many  thousand  feet  as  the  year 
Contains  days  (365),  and  each  second  is  equivalent  to  about  100  feet. 

GEOCENTRIC  PARALLAX. 

§  102.  The  bodies  of  the  solar  system  being  comparatively  near  to  the 
earth,  a  change  in  a  spectator's  place  on  the  earth's  surface  gives  to  them 
a  sensible  parallactic  motion  on  the  surface  of  the  celestial  sphere,  and 
two  observers  at  remote  stations  would  not  assign  to  these  bodies  the  same 
places  at  the  same  time  without  first  clearing  their  observed  co-ordinates 
of  this  source  of  discrepancy.  The  mode  of  correction  is  to  refer  all  obser- 
vations to  one  common  station,  and  this  station  is  assumed,  for  conve- 
nience, to  be  at  the  centre  of  the  earth. 

§  103.  The  place  in  which  a  body  would  appear,  if  viewed  from  the 
centre  of  the  earth,  is  called  its  Geocentric  Place. 

§  104.  The  apparent  change  of  a  body's  place  that  would  arise  from  a 
change  of  the  spectator's  station  from  the  surface  to  the  centre  of  tho 
earth  is  called  Geocentric  Parallax. 

§  105.  The  transfer  of  station  from  the  surface  to  the  centre  of  th<< 
earth  is  sensibly  in  a  vertical  circle,  and  the  geocentric  parallax  is  there- 
fore in  the  same  plane. 

§  106.  The  co-ordinates  of  a  body's  place,  as  determined  by  observa- 
tion, corrected  for  geocentric  parallax,  are  the  geocentric  co-ordinates  ol 
the  body. 

§  107.  The  point  in  which  the  radius  of  the  earth  produced  through 
the  spectator's  place  pierces  the  celestial  sphere,  is  called  the  central  zenith. 
The  arc.  of  the  celestial  meridian  from  the  central  zenith  to  the  equinoc 


GEOCENTRIC    PARALLAX. 


25 


Fig.  40. 


tial,  is  called  the  central  latitude.     The  difference  between  the  latitude 
and  the  central  latitude,  is  called  the  reduction  of  latitude. 

Thus  BAB'  A',  being  a  meridian 
section  of  the  terrestrial  spheroid,  and 
Z  Q  an  arc  of  the  celestial  sphere  in  the 
same  plane,  M  the  spectator's  place,  Q 
the  highest  point  of  the  equinoctial, 
MG  the  direction  of  the  plumb-line, 
CM  the  radius  of  the  earth  ;  then  will 
Z'  be  the  central  zenith,  Z'  Q  the  cen- 
tral latitude,  and  ZMZf  =  Z  Q—Z'Q, 
the  reduction  of  latitude. 

§  108.  Denote  in  future  the  central 
latitude  by  /,  the  polar  radius  by  y,  and  the  latitude  by  /',  then,  Appen- 
dix No.  IV.,  equation  ((?), 

tan£  =  /tan  /'       ........     (18) 

that  is,  the  tangent  of  the  central  latitude  is  equal  to  the  tangent  of  the 
latitude  into  the  square  of  the  polar  radius. 

Denote  the  radius  of  the  earth  drawn  to  the  spectator's  place  by  p, 
then,  Appendix  No.  IV.,  equation  (r), 


~ 
r 


(19) 


sin2/ 


Thus,  the  latitude  being  found  from  observation  (§  90),  the  centra, 
latitude  becomes  known  from  equation  (18),  and  hence  the  radius  of  th« 
earth  drawn  to  the  spectator's  place,  equation  (19). 

§  109.  Let  AB'A'B  be  a  meridian 
section  of  the  earth's  surface,  A  A  the 
equatorial  diameter,  Ml  and  Mz  the 
places  of  two  observers  viewing  the  same 
body  S.  The  observer  at  Mt  would  see 
the  body  projected  upon  the  celestial 
sphere  at  Si,  that  at  M^  would  see  it 
projected  at  £2,  and  to  an  observer  at 
the  centre  it  would  appear  at  S3.  The 
points  Z,  and  Z2  are  the  central  zeniths 
of  the  two  observers ;  Z,  £,  and  Z2  £2 

are  the  central  zenith  distances  of  S,  as  viewed  from  M{  and  Mt  respec- 
tively.    The  first  diminished  by  S3  Sl  and  the  second  bv  S3  Sa,  will  give 


SPHERICAL    ASTRONOMY. 


Zl  S3  and  Z2  S3  the  central  zenith  dis-  Fig.  41  bis. 

tances  as  they  would  appear  from  the 

centre.     But  S3  Sl  measures  the  angle 

S3  S  Si  =  MI  S  C,  and  S%  S3  the  angle 

S2  S  S3  =  M2  S  C ;  so  that  the  angles 

M  S  (7  and  M2  S  C  are  the  corrections 

for  parallax. 

§  110.  The  parallax  of  a  body  is  the 
angle  at  the  body  subtended  by  the 
earth's  radius  drawn  to  the  spectator. 

When  the  body  is  above  the  horizon, 

or  is  in  altitude,  it  is  called  the  parallax  in  altitude.     When  the  body  is 
in  the  horizon,  it  is  called  the  horizontal  parallax. 

Make 

z,  =  MI  S  C  =  parallax  in  altitude  at  Ml ; 

z2  =  M2  S  C  =  parallax  in  altitude  at  M2 ; 
P{  =  horizontal  parallax  at  Ml ; 
P2  =  horizontal  parallax  at  M2 ; 

r  =  C  S      =  distance  of  the  body  from  the  earth's  centre ; 

p!  =  MI  C    =  radius  of  the  earth  for  J\f{ ; 

p2  =  M2  C    =  radius  of  the  earth  for  M2 ; 
Zi  =  Z{  S{     =  central  zenith  distance  at  M \ ; 
Zj  =  Z2  S%     =  central  zenith  distance  at  M% : 

Z,  =  Z,  G  A  =  central  latitude  of  Ml ; 

£2  =  ZZCA=  central  latitude  of  M2. 

Then,  in  the  triangle  Mv  S  (7, 


whence 


sn  Z    :  sn 


sin  2,  =  —  .  sin  Zl ; 


:  r 


But  because  z{  is  always  very  small,  we  may  write 

z\ 

sin  2,  =  —  ; 
u 

in  which  u  is  the  number  of  seconds  in  radius,  and  zl  is  expressed  in  the 
•arae  unit ;  which  substituted  above  gives 

gl  =  w.^.sinZi; 


GEOCENTRIC    PARALLAX.  27 

when  Zt  becomes  90°,  the  body  is  in  the  horizon,  and  z,  becomes  P,,  and 
we  have 

PI  =  W.£    .........     (20) 

and  this  above  gives 

zl=Pl.sinZl    ........     (21) 

Whence  the  parallax  in  altitude  is  equal  to  the  horizontal  parallax  into 
the  sine  of  the  central  zenith  distance. 

§  111.  If  the  observer  be  upon  the  equator,  then  will  pi  become  unity, 
PI  becomes  the  horizontal  parallax  on  the  equator,  called  the  equatorial 
horizontal  parallax  ;  designating  this  latter  by  P,  we  have,  equation  (20), 


and  this  in  equation  (20)  gives 

Pi  =  P.pi       ........     (22) 

that  is  to  say,  the  horizontal  parallax  of  a  body  at  any  place,  is  equal  to 
the  product  of  the  equatorial  horizontal  parallax  of  the  body  by  the  ra- 
dius of  the  earth  at  the  place. 

The  value  of  P,  in  equation  (21)  gives 

zj  =  P  .  Pi  .  sin  Z,        ......     (23) 

§  112.  To  find   the   equatorial  horizontal  parallax  of  any  body,  we 
have  in  the  triangles  Ml  S  C  and  M^  S  C 


adding 

*i  +  *2  =  p-  (Pi  •  sin  zi  +  P,  -  sin 
but 


by  addition 

z,  -f  z2  =Z,  +  Z2  -  (Z,  <7S  +  Z2  <7S)  =  Z,  +  Z,  -  (/,  +  /.), 
which  substituting  above,  and  dividing  by  the  coefficient  of  P,  gives 

P  =  Zj  +  Zi-ft  +  i,) 

p,  sin  Z{  +  p2  sin  Z2 

§  113.  If  the  body  be  so  remote  that  the  difference  between  the  radii 
of  the  earth,  as  viewed  from  it,  be  insignificant,  which  is  the  ?ase  with  all 


28  SPHERICAL    ASTRONOMY. 

bodies  except  the  moon,  p,  and  p2  may  be  regarded  as  equal  to  oue  an- 
other, and  each  equal  to  unity,  and  we  shall  have,  equation  (24), 

, 


sin  Zi  +  sin  Z8 

in  which  /,  and  /2  are  the  central  latitudes  of  the  places  Ml  and  M^ 

§  114.  In  all  this  the  observers  have  been  supposed  to  be  on  the  same 
meridian  ;  but  this  is  not  necessary,  nor  would  it,  in  general,  be  the  case 
in  practice.  If  on  different  meridians,  make 

£  =  change  of  meridian  zenith  distance  of  the  body  in  the  interval 

between  two  consecutive  culminations  ; 

X  =  difference  of  longitude  of  the  two  observers,  expressed  in  time  ; 
8'  =  change  in  meridian  zenith  distance  while  passing  from  the  first 

to  the  second  meridian  ; 
then 

24h  :  $  :  :  X  :  5'  ; 
whence 


If  the  meridian  zenith  distance  be  increasing  at  the  easterly  station,  o' 
is  to  be  added  to,  if  decreasing,  subtracted  from,  the  meridian  zenith  dis- 
tance at  that  station.  This  corrected  meridian  zenith  distance  will  be 
that  which  the  body  would  have  to  an  observer  on  the  meridian  of  the 
westerly  station,  and  on  the  same  parallel  of  latitude  with  the  observer 
on  the  easterly  meridian,  the  reduction  being  in  eifect  to  bring  the  ob- 
servers to  the  same  meridian. 

§  115.  To  recapitulate:  the  latitudes  of  two  stations  are  first  found 
from  observation  ;  the  central  latitudes  are  found  from  equation  (18)  ;  the 
radii  of  the  earth  at  the  two  stations,  from  equation  (19)  ;  the  equatorial 
horizontal  parallax,  from  equation  (24)  ;  the  horizontal  parallax  at  any 
place,  from  equation  (22)  ;  and  the  parallax  in  altitude,  from  equation  (23), 

AUGMENTED  AND  HORIZONTAL  DIAMETERS. 

§  116.  By  the  rotation  of  the  earth  upon  its  axis  the  spectator  is  con- 
tinually changing  his  distance  from  the  heavenly  bodies.  A  change  of  dis- 
tance gives  rise  to  a  change  in  the  apparent  dimensions  of  an  object.  A 
body  seen  in  the  horizon  of  a  spectator  would  appear  to  him  sensibly  of 
the  same  size  as  if  seen  from  the  centre  of  the  earth,  the  distances  If  C 
and  H  M,  for  the  nearest  of  the  heavenly  bodies,  being  sensibly  the  same. 


DIMENSIONS    OF   THE    HEAVENLY   BODIES. 


29 


The  apparent  semi-diameter  of  a  body 
is  the  angle  at  the  observer  subtended  by 
the  body's  real  serai-diameter,  the  latter 
being  perpendicular  to  a  visual  ray  drawn 
to  one  of  its  extemities. 

Make 

y  =  HM B  =  apparent  semi-diameter  of 

a  body  when  in  the  horizon ; 
6''  =  L  M B'=  apparent  semi-diameter  of 

the  body  when  in  altitude ; 
d  =  HB  =  L 13'=  real  semi-diameter  of  the  body  in  linear  units; 
r  =  body's  distance  from  the  observer  when  in  the  horiztn  =  distance 

from  earth's  centre ; 

r '  =  body's  distance  from  the  spectator  when  in  altitude  ; 
Z  —  Z' ML  —  the  body's  central  zenith  distance; 
z  =  ML  C  =  the  body's  parallax  in  altitude.  , 

Then 

r  .  sin  s  =  d  =  r7  sin  sf ; 
whence 


sin  sr 
sin  s 


sin  Z 


1 


sin  (Z-z) 


cos  z 


cos  Z 
sin  Z 


sin  z 


replacing  sin  z  by  its  value  — ,  and  z  by  its  value  in  Eq.  (23),  also  writing 

-  for  sin  s,  and  -  for  sin  s'  we  find 
u  to 

sf  1 

....     (27) 


COS  Z  —  p,- •  COS  Z 

u 


in  which  s1  and  s  are  expressed  in  seconds  of  arc. 

§  117.  The  apparent  diameter  2s  of  a  body  in  the  hon'zon,  is  called  the 
horizontal  diameter ;  its  apparent  diameter  2  sf  in  altitude,  is  called  the 
augmented  diameter. 

DISTANCES  AND  DIMENSIONS  OF  THE  HEAVENLY  BODIES. 

§  118.  Having  found  the  horizontal  parallax  of  a  body,  it  is  easy  to  find 
its  distance  from  the  earth's  centre.  From  equation  (20)  we  have 


(28) 


in  which  p  $nd  P  are  respectively  the  equatorial  radius  of  the  earth  and 


30  SPHERICAL    ASTRONOMY. 

the  equatorial  horizontal  parallax  of  the  body ;  and  from  which  we  con- 
clude that  the  distance  of  any  body  from  the  earth's  centre,  is  equal  to  the 
equatorial  radius  of  the  earth  repeated  as  many  times  as  the  number  of 
seconds  in  the  body's  equatorial  horizontal  parallax  is  contained  in  the 
number  of  seconds  in  radius. 

§  119.  The  horizontal  parallax  of  a  body  is  the  apparent  semi-diameter 
of  the  earth  as  seen  from  the  body.  The  apparent  semi-diameter  of  two 
bodies  seen  at  the  same  distance  are  directly  proportional  to  their  real 
magnitudes.  Make 

s  =  apparent  semi-diameter  of  the  body  ; 

d  =  the  real  semi-diameter  of  the  body  in  linear  units,  as  miles  I 

P=  the  equatorial  horizontal  parallax  of  the  body  ; 

p  =  the  equatorial  radius  of  the  earth  ; 

then  will 

P  :  s  :  :  p  :  c?; 
whence 

d=f.j (29) 

that  is,  the  real  semi-diameter  of  any  heavenly  body  is  equal  to  the  equa- 
torial radius  of  the  earth  repeated  as  many  times  as  the  body's  equatorial 
horizontal  parallax  is  contained  in  its  apparent  semi-diameter.  The  appa- 
rent diameter  of  a  body  is  measured  by  means  of  the  micrometer. 

ECLIPTIC. 

§  120.  The  orbit  of  the  earth  about  the  sun  is  sensibly  a  plane  curve. 
The  intersection  of  the  plane  of  the  earth's  orbit  with  the  celestial  sphere 
is  called  the  ecliptic.  The  ecliptic  is  a  great  circle  of  the  celestial  sphere 
because  its  plane  passes  through  the  earth's  centre. 

§  121.  The  orbital  motion  of  the  earth  about  the  sun  gives  rise  to  a 
parallactic  motion  of  the  sun  about  the  earth,  and  the  effect  to  a  spectator 
on  the  earth  is  the  same  as  though  the  latter  were  stationary  and  the  sun 
in  motion  about  the  earth.  The  sun  appears  to  move  along  the  ecliptic  in 
the  same  direction  that  the  earth's  projection  upon  the  celential  sphere,  as 
seen  from  the  sun,  actually  moves  in  that  great  circle. 

§  122.  The  earth's  axis  being  oblique  to  the  plane  of  the  ecliptic,  forms 
an  angle  with  the  radius  vector  of  the  earth.  The  axis  of  the  earth  re- 
taining its  direction  sensibly  unchanged,  this  angle  is  variable. 

§  123.  Let  S  be  the  sun,  EiEllElllEllll  the  earth's  orbit,  PS  a  line 
through  the  sun's  centre  and  Darallel  to  the  direction  of  the  earth's  axis. 


ECLIPTIC. 


31 


EtEul  the  projection  of  this  line  on  the  plane  of  the  ecliptic.  Draw 
EuSElul  perpendicular  to  EtEllfl  and  EtPfl  EnPlt,  EinPltl,  and 
EtlllPlin  parallel  to  8 P.  The  angle  PlEl S  will  be  the  polar  distance 
«f  the  sun  when  the  earth  is  at  Ef,  PltEtlS  when  at  E(l,  PulEinS 
when  at  EUI,  and  PnilElulS  when  at  Elllt.  It  will  be  least  at  Et, 
greatest  at  EIU,  and  90°  at  Eu  and  EIIU.  The  polar  distance  at  E,,,  is 
the  supplement  of  that  at  E,,  and  estimated  from  the  nearest  or  opposite 
poles,  the  polar  distances  at  E,  and  E///  are  equal. 

§  1 24.  The  declination  being  the  complement  of  the  polar  distance,  the 
sun's  declination  will  sometimes  be  north,  sometimes  south.  Its  north  de- 
clination will  be  greatest  when  its  north  polar  distance  is  least ;  its  south 
declination  greatest  when  its  south  polar  distance  is  least. 

§  125.  Thus,  by  the  orbital  motion  of  the  earth,  the  terrestrial  equator 
is  carried  from  one  side  of  the  sun  to  the  opposite,  and  the  sun  itself  made 
apparently  to  pass  alternately  from  north  to  south  and  from  south  to  north 
of  the  equinoctial. 

§  126.  The  radius  vector  would,  by  the  orbital  motion  alone,  trace  upon 
the  surface  of  the  earth  an  ellipse  of  which  the  plane  would  coincide  with 
that  of  the  ecliptic ;  by  the  diurnal  motion  alone,  it  would  trace  a  parallel 
of  latitude ;  and  by  both  motions  combined,  it  actually  describes  a  kind  of 
spiral  curve  extending  on  either  side  of  the  equator  and  intersecting  all  the 
parallels  between  those  whose  latitudes  are  equal  to  the  sun's  greatest 
oorth  and  south  declinations.  To  spectators  situated  somewhere  on  thes« 


SPHERICAL    ASTRONOMY. 


parallels,  the  sun  will  be  vertical,  or  in  the  zenith,  twice  in  the  course  of 
one  revolution  of  the  earth  about  the  sun. 

§  127.  Two  small  circles  of  the  celestial  sphere  parallel  to  the  equinoc- 
tial and  at  a  distance  therefrom  equal  to  the  sun's  greatest  north  and  south 
declination  are  called  tropics ;  that  on  the  north  is  called  the  tropic  of 
Cancer,  arid  that  on  the  south  the  tropic  of  Capricorn. 

§  128.  A  plane  through  the  earth's  centre,  and  perpend Lcular  to  the 
radius  vector,  divides  the  earth's  surface  into  two  equal  parts.  That  on 
the  side  towards  the  sun  is  illuminated,  while  that  on  the  opposite  side  is 
in  the  dark.  To  an  observer  on  the  former  it  is  day  ;  to  one  on  the  latter 
it  is  night.  The  curve  which  separates  the  enlightened  portion  from  the 
dark  is  called  the  circle  of  illumination. 

§  129.  When  the  earth  is  in  either  of  the  positions  E,,  or  E,,,,,  its 
axis  is  in  the  plane  of  the  circle  of  illumination ;  this  latter  divides  all  par- 
allels equally,  and  the  lengths  of  the  days  are  equal  to  those  of  the  nights 
all  over  the  earth's  surface. 

When  the  earth  is  in  either  of  the  positions  E/  or  E,/n  its  axis  makes 
the  greatest  angle  possible  with  the  plane  of  the  circle  of  illumination  ;  the 
latter  divides  the  parallels  most  unequally,  and  the  length  of  the  days  will 
diffej  from  those  of  the  nights  the  most  possible. 

§  130.  To  all  places  north  of  the  parallel  whose  latitude  is  equal  to  the 
north  polar  distance  of  the  sun,  the  sun  will,  Eq.  (7),  be  circumpolar, 
while  to  all  places  having  an  equal  south  latitude  the  sun  will  not,  Eq.  (7), 


// 

ECLIPTIC. 


rise.     In  like  manner,  to  all  places  south  of  the  parallel  whose 

p^ual  1o  the  south  polar  distance  of  the  sun,  the  sun  will  be  circumpolar  *, 

while  to  all  places  of  equal  north  latitude,  the  sun  will  not  rise. 

§  131.  During  the  time  the  earth  is  moving  from  E,,  to  E///f,  the  sun 
will  shine  upon  the  south  pole,  and  the  north  pole  will  be  deprived  of  his 
direct  light.  While  moving  from  E,,,,  to  E,,  the  reverse  will  bo  true. 
The  zones  of  polar  illumination  and  obscuration  will  increase  from  E// 
to  E/f/  and  from  E/f//  to  E,,  and  diminish  from  E,  to  E,,  and  from  E//t 
to  E/t//.  The  radii  of  the  greatest  zones  of  polar  illumination  and  obscu- 
ration for  one  diurnal  revolution  are  P,I,  and  Pf//I//n  which  are  equal  to 
one  another,  being  the  greatest  departure  of  the  pole  from  the  circle  of  il- 
lumination on  opposite  sides. 

§  132.  Draw  E/  Q/  and  E///  Q///  respectively  perpendicular  to  P,E, 
and  Pt//Ef//\  then  will  Q/  Qt  and  Q'"  Qy//  represent  the  equator,  QlDi 
and  Q///D///  the  greatest  north  and  south  declinations  of  sun  respectively, 
and  Pflf  and  P UIIIU  the  greatest  departure  of  the  pole  from  the  circle  of 
illumination.  Now 

/,  D,  =  P,  Q,  =  90°=  /„, It,,,  =  P,,,  <?„„ 
P,».  =  P.-D.,  >.„•»,« -A**./. 

and  by  subtraction 

/>, /,  =  />,«„        PIIII.,,  =  Q.IIDI,I; 

that  is,  the  radius  of  the  zone  of  greatest  polar  illumination,  or  obscuration, 
is  equal  to  .the  greatest  declination  of  the  sun. 

§  133.  Two  small  circles  parallel  to  the  equinoctial,  and  at  a  distance 
from  the  poles  equal  to  the  greatest  declination  of  the  sun,  are  called  polar 
circles  ;  that  about  the  north  pole  is  called  the  arctic,  and  that  about  the 
south  the  antarctic  circle.  The  polar  circles  are  the  boundaries  of  the 
greatest  zones  of  polar  diurnal  illumination  and  obscuration. 

§  134.  When  the  intervals  of  time  between  three  consecutive  passages 
of  a  circumpolar  star  over  the  line  of  collimation  of  a  transit  or  mural  cir- 
cle are  equal,  these  instruments  are  adjusted  to  the  meridian. 

§  135.  The  diurnal  motion  brings  the  meridian  of  a  place,  in  the  course 
of  one  revolution  of  the  earth  on  its  axis,  into  coincidence  with  the  decli- 
nation circle  of  every  body  in  the  heavens.  The  difference  of  times  between 
the  meridian's  passing  the  centres  of  any  two  bodies,  is  the  difference  of 
right  ascension  of  these  bodies. 

§  136.  To  find  the  time  o*  the  meridian's  passing  the  centre  of  any 
body,  find  by  the  transit  instrument  and  timepiece  the  time  of  th<:  merid- 
ian's passing  the  body's  east  and  west  limb,  and  take  half  the  sum. 

3 


34.  SPHERICAL    ASTRONOMY. 

§  137.  To  find  the  polar  distance  of  a  body's  centre,  take  the  reading  of 
the  mural  circle  when  its  line  of  collirnation  is  upon  the  upper  or  lower 
lirnb ;  subtract  from  this  the  polar  reading  and  correct  the  difference  for 
refraction,  parallax  in  altitude,  and  semi-diameter.  The  declination  is  ob- 
tained by  subtracting  the  polar  distance  from  90°. 

§  138.  The  points  in  which  the  equinoctial  intersects  the  ecliptic  are 
called  the  equinoxes  ;  that  by  which  the  sun  passes  from  the  south  to  the 
north  of  the  equinoctial  is  called  the  vernal  equinox  ;  the  other,  or  that  by 
which  the  sun  passes  from  the  north  to  the  south  of  the  equinoctial,  is  called 
the  autumnal  equinox. 

§  139.  The  angle  which  the  equinoctial  makes  with  the  ecliptic  is  called 
the  obliquity  of  the  ecliptic. 

§  140.  To  find  the  place  of  the  vernal  equi- 
nox and  the  obliquity  of  the  ecliptic,  let  VDZ 
be  an  arc  of  the  equinoctial,  VS2  of  the  eclip- 
tic, V  the  vernal  equinox,  $,  and  Sz  two 
places  of  the  sun  when  on  the  meridian  at 
different  times,  £,/>,,  SZD2  arcs  of  declina- 
tion circles ;  and  make 

<£,  =  D{  $b  the  sun's  declination  at  any  meridian  passage  ; 
#2  =  D2  S&  the  same  at  some  subsequent  passage ; 
2a  =  VD2  —  VD{,  the  corresponding  difference  of  right  ascension  ; 
x  =  VDi,  the  right  ascension  of  the  sun  at  the  time  of  first  meri<han 

passage ; 
w  =  Si  VD\,  the  obliquity  of  the  ecliptic. 

Then  in  the  triangles  £t  VD{  and  S2  FZ>2,  right-angled  at  D{  and  D2l 

sin  x  —  tan  #,     cot  w, 
sin  (x  -f-  2a)  =  tan  <J2  .  cot  w  ; 
.•and  by  division 

sin  (x  +  2a)  =  tan  S2  ^ 
sin  x  tan  ^^ 

adding  unity  and  clearing  the  fraction,  then  subtracting  unity  ard  clear 
ing,  and  dividing  one  result  by  the  other,  we  find 

sin  (x  -f-  2a)  +  sin  x      tan  £8  -f-  tan  5, 
sin  (x  -f  2a)  —  sin  x      tan  #f  —  tan  ^  ' 

tan  (x  +  a)  _  sin  (Ss  -f-  5,) 
tan  a  sin  (6S  —  5,)' 


ECLIPTIC.  35 

Whence 

to  (*  +  . )=*§+*!>    tana.    ...        (31) 
sm  (£2  —  £t) 

Also 

cot  w  =  sin  x  .  cot  <5, (32) 

The  value  of  the  obliquity  is  thus  found  to  be  nearly  23°  27'  54",  which 
is  therefore  the  greatest  north  and  south  declination  of  the  sun.  The  tropics 
are,  therefore,  23°  27'  54"  from  the  equinoctial,  and  the  polar  circles  are 
at  the  same  distance  from  the  poles. 

§  141.  The  interval  of  time  between  the  sun  and  a  star  crossing  the 
meridian,  applied  to  the  right  ascension  of  the  sun,  gives  the  right  ascen- 
:»ion  of  the  star.  The  declination  of  a  star  is  found  like  that  of  the  sun, 
except  that  there  is  no  correction  for  parallax  and  semi-diameter,  the  only 
correction  being  for  refraction. 

§  142.  The  right  ascension  and  declination  of  one  star  being  known, 
the  differences  of  observed  right  ascensions  and  declinations,  the  latter  being 
corrected  for  differences  of  refractions,  give,  when  applied  to  the  right  as- 
cension and  declination  of  the  known  star,  the  right  ascension  and  decli- 
nation of  other  stars.  Thus  a  list  of  the  stars,  together  with  their  right 
ascensions  and  declinations,  and  arranged  in  the  order  of  their  right  ascen- 
sions, furnishes  the  ground-work  of  what  is  called  a  catalogue  of  stars,  of 
vvhich  a  fuller  account  will  be  given  presently. 

§  143.  A  belt  of  the  heavens  extending  on  either  side  of  the  ecliptic, 
far  enough  to  embrace  the  paths  of  the  planets,  is  called  the  zodiac. 

§  144.  The  ecliptic  is  divided  into  twelve  equal  parts,  called  signs. 
They  commence  at  the  vernal  equinox,  and  are  named  in  order,  proceed- 
ing towards  the  east,  Aries  (T),  Taurus  (»),  Gemini  (n),  Cancer  (£?), 
Leo  (ty),  Virgo  ("HE),  Libra  (===),  Scorpio  (^),  Sagittarius  (  #  ),  Capricor- 
nus  (V?),  Aquarius  (~),  and  Pisces  (X).  Motion  in  the  order  of  the 
signs  is  said  to  be  direct ;  the  converse,  retrograde. 

§  145.  The  points  of  the  ecliptic  in  which  the  sun  reaches  his  greatest 
'north  and  south  declination  are  called  the  solstitial  points :  that  on  the 
north  is  called  the  summer  solstice,  and  that  on  the  south  the  winter  sol- 
stice. The  sun  when  in  these  points  appears  to  be  stationary  as  regards  his 
apparent  motion  in  declination.  The  solstitial  colure  is  the  declination 
circle  through  the  solstitial  points.  The  equinoctial  colure  is  the  declina- 
tion circle  through  the  equinoctial  points.  The  solstitial  colure  separates 
Gemini  from  Cancer,  and  Sagittarius  from  Capricornus;  the  equinoctial 
colure  separates  Aries  from  Pisces,  and  Virgo  from  Libra. 

§  146.  A  great  circle  of  the  celestial  sphere  passing  through  the  pole> 
of  the  ecliptic  is  called  a  circle  of  latitude. 


SPHERICAL    ASTRONOMY. 


g  147.  The  latitude  of  a  body  is  the  distance  of  the  bouy's  AH  tie  from 
the  ecliptic,  measured  on  a  circle  of  latitude. 

§  148.  The  longitude  of  a  body  is  the  distance  from  the  vernal  equinox 
to  the  circle  of  latitude  through  the  body's  centre,  measured  on  the  eclip- 
tic in  the  order  of  the  signs. 

The  longitude  and  latitude  are  co-ordinates  that  refer  a  body's  place  to 
the  circle  of  latitude  through  the  vernal  equinox  and  to  the  ecliptic  ;  the 
longitude  and  ecliptic  polar  distance  are  polar  co-ordinates  that  refer  a 
body's  place  to  the  same  circle  of  latitude  and  to  the  pole  of  the  ecliptic- 

§  149.  The  longitude  of  the  sun,  as  seen  from  the  earth,  is  readily  ob- 
tained from  the  obliquity  of  the  ecliptic  and  either  the  right  ascension  or 
declination. 

For  this  purpose  make  Fi°-  u  bis- 

a  =:  VS^  the  longitude  of  the  sun; 

8  —  SiDn  his  declination; 

a  =  VDi,  his  right  ascension ; 

w  =  S{  VDh  the  obliquity  of  the  ecliptic. 


Then  will 


tan  a  = 


sin  «  = 


tan  a 

COS  W 

sin  5 


(33) 


(34) 


§150.  The  place  of  the  sun  as  seen  from  the  earth,  and  that  of  the 
earth  as  seen  from  the  sun,  are  at  the  opposite  extremities  of  the  same  di- 
ameter of  the  ecliptic;  and  the  longitude  of  the  sun,  increased  by  180°, 
will  be  the  longitude  of  the  earth  as  viewed  from  the  sun,  the  centre  of  the 
earth's  orbital  motion. 

§  151.  The  sun  appears  in  the  vernal  equinox  on  the  20th  March,  in 
the  autumnal  equinox  on  the  22d  September,  the  summer  solstice  on  the 
21st  June,  and  in  the  winter  solstice  on  the  21st  December. 

The  poles  of  the  ecliptic  are  at  a  distance  from  the  nearest  poles  of  the 
equinoctial,  equal  to  the  obliquity  of  the  ecliptic. 

g  152.  The  right  ascension  is  obtained  from  observation  by  means  of 
the  clock  and  transit  instrument,  the  declination  by  means  of  the  mural 
circle.  From  these  and  the  obliquity  of  the  ecliptic,  the  longitude  and 
latitude  are  obtained  from  computation.  Thus,  let  S  be  the  body's  place, 
V  the  vernal  equinox,  VD  the  body's  right  ascension,  D  S  its  declina- 


PRECESSION    AND    NUTATION. 


tion,  VL  its  longitude,  S  L  its  latitude,  and  the  *"*«• 

angle  L  VD  the  obliquity  of  the  ecliptic. 

Make  //   \ 

a  =  VD  =  right  ascension ;  ^z^/L          \<S 

£  =.  D  S  =  declination ;  /          ,-K     \ 

f  =  VL  —  longitude ;  ~~^&^\*>    \  \ 

X  =  L  S  =  latitude ; 
w  =  L  VD  =  obliquity  of  the  ecliptic ; 
e  =  S  V  =  arc  of  great  circle  through  the 

body  and  vernal  equinox ; 
9  =  S  VD  =  inclination  of  S  V  to  the  equinoctial 

Then  in  the  triangle  S  VD,  right-angled  at  D, 

tan  5 

ten  9  = (35) 

sin  a 

cos  e  =  cos  a  .  cos  § (36) 

and  in  the  triangle  VL  S,  right-angled  at  Z, 

tan  I  =  tan  e  .  cos  (9  —  00) (37) 

sin  X  =  sin  s  .  sin  (9 — w) (38) 

§   153.  If  the  longitude,  latitude,  and  obliquity  be  given,  then  in  the 
triangle  VLS, 

tan  X 
tan  (9  —  w)  =  -r-j (39) 

cos  e  =  cos  /  .  cos  X (40) 

and  in  the  triangle  VD  S, 

tan  a  =  tan  s  .  cos  9 (41) 

sin  5  =  sin  s  .  sin  9       ......     (42) 


PRECESSION  AND  NUTATION. 

§  154.  The  longitudes  and  latitudes  of  the  stars,  being  thus  determined 
at  different  epochs,  show  a  slow  increase  in  all  the  longitudes,  while  the 
latitudes  remain  sensibly  the  same. 

§  155.  This  is  owing  to  a  slow  gyratory  motion  of  the  line  of  the  ter- 
restrial poles  in  a  retrograde  direction,  caused  by  the  rotary  motion  of 
the  earth  and  the  combined  action  of  the  sun,  moon,  and  planets  upon 
the  ring  of  equatorial  matter  that  projects  beyond  the  sphere  of  which  the 
polar  axis  is  the  diameter.  It  is  the  resultant  of  two  component  motions. 


SPHERICAL    ASTRONOMY. 


Fig.  4a 


§  156.  By  the  first  of  these  components  alone,  called  nutation,  the 
line  of  the  poles  would  describe  once  in  every  19  years  an  acute  conical 
surface,  of  which  the  vertex  is  at  the  centre  of  the  earth,  and  the  intersec- 
tions with  the  celestial  sphere  are  two  equal  ellipses,  whose  transverse  and 
conjugate  axes  are  respectively  18". 5  and  13".74,  the  former  being  al- 
ways directed  towards  the  poles  of  the  ecliptic 

§  157.  By  the  second,  called  the  mean  precession,  the  centres  of  these 
ellipses  are  carried  uniformly  around  the  poles  of  the  ecliptic  from  east  to 
west  in  equal  circles,  of  which  the  radii  are  about  23°  28',  and  at  a  rate 
of  50 ".2  in  the  interval  of  time  between  two  consecutive  returns  of  the 
sun  to  the  mean  vernal  equinox.  This  interval  is  called  a  tropical 
year.  The  mean  equinoxes  perform,  therefore,  one  entire  revolution  in 
360°  4-  50".2  =  25817  tropical  years. 

In  the  figure,  S  is  the  sun,  E  the  earth,  Pt  the  north  pole  of  the  ecliptic, 
P'  the  true  and  P  the  mean  north  pole  of  the  equinoctial.  The  eurva 
about  S  represents  the  earth's  orbit,  that 
about  E,  and  of  which  the  plane  is  perpen- 
dicular to  E  P',  shows  the  direction  of  the 
earth's  axial  motion  ;  E  V  is  the  intersection 
of  the  plane  of  this  circle  with  that  of  the 
ecliptic,  and  Fis  the  vernal"  equinox.  The 
circle  about  P,  has  a  radius  of  about  23°  28', 
the  curve  about  P  is  the  elliptical  path  de- 
scribed by  the  true  pole  Pf  about  the  mean 
P,  and  of  which  the  longer  axis  P'P"  passes 
through  Pr  The  arc  V L  of  the  ecliptic, 
the  circle  about  P,,  and  ellipses  about  P  are 
all  on  the  surface  of  the  celestial  sphere,  while 
them,  are  at  its  centre. 

§  158.  By  the  component  motions  of  mean  precession  and  of  nutation 
combined,  the  true  equinoctial  pole  is.  carried  with  a  variable  motion  along 
a  gently  waving  curve  whose  undulations  extend  to 
equal  distances  on  either  side  of  the  circumference 
of  mean  precession,  and  which  it  intersects  at  points 
separated  by  angular  distances,  as  seen  from  the 
centre  of  the  celestial  sphere,  equal  to  19  X  50".2 
X sin  23°  28'-=-  2  ±  13",74  =  3'  10"  ±  18",74. 

§  159.  The  motions  due  to  the  action  of  the  sun 
afld  moon  are  opposed  to  those  arising  from  the  ac- 
tion of  the  planets,  and  when  estimated  along  the 


E,  and  the  curves  about 


Fig.  47. 


PRECESSION    AND    NUTATION.  $Q 

ecliptic  are  called  luni-solar  precession  in  longitude.  The  combined  effect 
aii sing  from  the  simultaneous  action  of  all  the  bodies,  estimated  in  the 
same  direction,  is  called  the  general  precession  in  longitude. 

§  160.  The  equinoxes  always  conforming  to  the  places  of  the  equinoc- 
tial poles,  have  a  slow,  irregular,  but  continuous  retrograde  motion. 

The  place  of  the  vernal  equinox  without  nutation  is  called  the  mean, 
equinox ;  with  nutation,  the  true  or  apparent  equinox. 

The  inclination  of  the  equinoctial  to  the  ecliptic  without  nutation  is 
called  the  mean  obliquity ;  with  nutation,  the  true  or  apparent  obliquity. 
The  difference  between  the  mean  and  apparent  obliquity  is  called  the  nu- 
tation of  obliquity. 

§  161.  The  apparent  equinox  wanders  in  either  direction  from  the 
mean  to  a  distance  equal  to  13", 74-^2  •  sin  23°  28',=  17",  25,  which  it 
reaches  when  the  mean  and  apparent  obliquity  are  equal  ;  and  the 
apparent  obliquity  varies  on  either  side  of  the  mean  from  zero  to  half 
of  18x/.5  or  9/x.25;  the  latter  being  reached  when  the  apparent  equinox 
coincides  with  the  mean. 

§  162.  The  motion  of  the  mean  equinox  along  the  ecliptic  is  deter- 
mined by  that  of  the  centre  of  the  little  ellipse  above  referred  to,  and  is 
therefore  at  the  rate  of  50".2  a  year,  being  the  quotient  which  results  from 
dividing  360°  by  the  period  required  for  the  true  pole  to  perform  one 
entire  circuit  around  the  pole  of  the  ecliptic. 

§  163.  The  distance  from  the  mean  to  the  apparent  equinox  is  called 
the  equation  of  the  equinoxes  in  longitude. 

§  164.  The  intersection  of  a  declination  circle  through  the  mean  equi- 
nox with  the  equinoctial,  is  called  the  reduced  place  of  the  mean  equinox. 

§  165.  The  distance  from  the  reduced  place  of  the  mean  equinox  to 
the  apparent  equinox,  is  called  the  equation  of  the  equinoxes  in  right  as- 
cension. 

§  166.  The  changes  which  take  place  in  these  equations,  as  also  in  the 
apparent  obliquity  of  the  ecliptic,  are  called  periodical  variations,  from  the 
circumstance  of  their  running  through  all  their  possible  values  in  a  com- 
paratively short  period. 

Formulas  for  computing  the  equations  of  the  equinoxes  in  longitude 
and  right  ascension  will  be  given  in  another  place. 

§  167.  Besides  the  motion  of  the  equinoctial,  due  to  the  action  of  the 
heavenly  bodies  on  the  protuberant  ring  of  matter  about  the  terrestrial 
equator,  there  is  another  effect  due  to  the  deflecting  action  of  the  planets-. 
By  this  the  earth  is  turned  aside  from  the  path  it  would  describe,  if  subject- 
ed to  the  action  of  the  sun  alone,  and  the  place  of  the  ecliptic,  therefore, 

7 


40  SPHERICAL    ASTRONOMF. 

changed.  The  amount  of  this  change  is  exceedingly  smal  ,  being  only 
about  46"  in  a  century.  Its  present  effect  is  to  diminish  the  mean  obli- 
quity, and  this  will  continue  to  be  the  case  for  a  long  period  of  ages,  when 
the  change  will  be  in  the  opposite  direction,  the  motion  being  one  of  oscil- 
lation to  the  extent  of  1°  21'  about  a  mean  position.  The  change  in  the 
value  of  the  mean  obliquity  arising  from  the  cause  here  referred  to,  is 
called  the  secular  variation  of  the  obliquity,  because  of  the  great  period 
of  time  required  to  pass  through  all  its  values. 

SIDEREAL  TIME. 

§  168.  It  has  been  explained  (p.  237)  how  the  motion  of  the  pointers  or 
hands  of  clocks  and  watches  over  stationary  circular  scales  of  equal  parts 
upon  their  dial-plates,  is  employed  to  measure  the  lapse  of  time.  The  uni- 
form motion  of  the  meridian,  carrying  with  it  an  imaginary  movable  circu- 
lar scale  of  equal  parts,  coincident  with  ^he  equinoctial,  gives  the  means 
of  regulating  these  and  all  other  artificial  time-keepers. 

§  169.  The  origin  or  zero  of  the  equinoctial  scale  is  on  the  upper  me- 
ridian; its  unit  of  measure  is  one  hour,  equal  to  15° ;  its  pointer  or  hatid 
the  declination  circle  through  the  centre  of  some  heavenly  body,' and  time 
measured  upon  it  takes  the  name  of  the  body  which  regulates  the  pointer. 

§  170.  The  distance  of  the  pointer  from  the  origin  or  upper  meridian, 
estimated  westwardly,  is  the  hour  angle  of  the  body  which  gives  the  scale 
its  name,  and  measures  the  time  since  its  meridian  passage. 

§  171.  Time  measured  by  the  hour  angle  of  the  mean  equinox  is 
called  mean  sidereal  time  ;  and  the  interval  of  time  between  two  consecu- 
tive passages  of  the  meridian  over  the  mean  equinox,  is  called  a  sidereal  day. 

§  172.  Time  measured  by  the  hour  angle  of  the  apparent  equinox  is 
called  apparent  sidereal  time  ;  and  the  interval  of  time  between  two  con- 
secutive passages  of  the  meridian  over  the  apparent  equinox,  is  called  an 
apparent  sidereal  day. 

§  173.  Apparent  sidereal  time  is  that  usually  employed  by  astronomers. 
It  is  affected  by  the  equation  of  the  equinoxes  in  right  ascension,  of  which 
the  value  in  time  being  applied  to  the  apparent  sidereal  time,  with  its 
proper  sign,  gives  the  mean  sidereal  time.  This  difference  between  appz.  • 
rent  and  mean  sidereal  time  is  called  also  the  equation  of  sidereal  time. 

§  174.  Apparent  sidereal  days  are  slightly  unequal;  but  the  fluctua- 
tions of  a  clock  marking  apparent  from  one  noting  mean  sidereal  time 
would  be  only  about  2*.3  in  nineteen  years. 

§  175.  A  timepiece  whose  hour-hand  passes  unif>rmly  over  the  circular 


THE    EARTH'S    ORBIT  41 

scale  of  24  hours  on  the  dial  -plate,  in  a  sidereal  day,  is  said  to  run  with 
sidereal  time  ;  it  will  mark  mean  sidereal  time  when  its  hands  indicate  at 
any  and  every  instant  the  hour  angle  of  the  mean  vernal  equinox. 

§  176.  The  sidereal  time  of  the  meridian's  passing  the  centre  of  any 
body  is  the  true  right  ascension  of  the  body;  and  the  rate  of  the  time- 
piece on  sidereal  time,  its  error  at  any  epoch,  and  the  indication  of  the 
hands  on  its  dial-plate  at  the  instant  the  meridian  passes  the  centre  of  any 
body,  are  the  data  which  make  known  the  body's  right  ascension. 

§  177.  The  sidereal  day  is  shorter  than  the  time  required  for  the  earth 
to  turn  once  about  its  axis  by  about  TJ^  of  a  sidereal  second. 

THE  EARTH'S  ORBIT. 

§  178.  The  orbit  of  the  earth  is  an  ellipse,  of  which  the  sun  occupies 
one  of  the  foci. 

§  179.  The  extremities  of  the  transverse  axis  of  the  orbit  are  called 
the  Apsides  ;  that  most  remote  from  the  sun  is  called  the  higher  and  that 
nearest  to  the  sun  the  lower  apsis.  The  lower  apsis  is  also  called  the  pe- 
rihelion and  the  higher  apsis  the  aphelion.  The  transverse  axis  produced 
both  ways  is  called  the  line  of  the  apsides. 

§  180.  The  place  of  the  sun  or  other  heavenly  body  which  has  the 
greatest  distance  from  the  earth  is  called  the  apogee,  and  that  which  has 
the  least  distance  is  called  the  perigee.  When,  therefore,  the  earth  is  in 
aphelion,  the  sun  is  in  apogee  ;  and  when  the  earth  is  in  perihelion,  the 
sun  is  in  perigee. 

§  181.  The  quotient  obtained  from  dividing  the  circumference  of  a 
circle,  of  which  the  radius  is  unity,  by  the  interval  of  time  between  two 
consecutive  returns  of  a  body  to  the  same  origin,  is  called  the  body's  mean 
motion  from  that  origin. 

Thus,  let  T  be  the  interval,  and  m  the  mean  motion  ;  then  will 


§  182.  The  origin  may  be  movable  or  fixed  ;  when  in  motion,  the  mo- 
tion may  be  direct  or  retrograde. 

§  183.  Denote  by  r  the  radius  vector  of  the  earth,  by^c  the  area  which 
this  line  describes  in  a  unit  of  time,  and  by  n  the  true  motion,  tl  en  will, 
Analytical  Mechanics,  equation  (266), 

»,=  2      .........          (44) 


SPHERICAL    ASTRONOMY. 


§  184.  The  interval  of  time  between  two  consecutive  returns  of  the 
sun  to  the  vernal  equinox,  is  called  a  tropical  year.  That  between  two 
consecutive  returns  to  the  mean  vernal  equinox,  a  mean  tropical  year. 

§  185.  The  arc  of  the  ecliptic  from  the  mean  vernal  equinox  to  the 
place  the  sun  would  occupy,  had  his  motion  in  longitude  been  uniform 
and  equal  to  a  mean  of  his  actual  motions,  is  called  his  mean  longi- 
tude— the  true  and  mean  places  always  coming  together  on  the  line  of 
the  apsides 

§  186.  The  interval  of  time  between  two  consecutive  returns  of  the 
earth  to  the  perihelion  or  aphelion  is  called  an  anomalistic  year. 

§  187.  The  mean  motion  of  the  earth  from  perihelion  is  the  value  of 
m,  in  equation  (43),  the  value  of  T  therein  being  the  anomalistic  year. 

§  188.  The  angle  E  S  P,  which  the  radius  vector  of  the  earth  makes  at 
any  time  with  the  line  of  the  apsides,  reckoned  from  perihelion,  is  called 
the  true  anomaly. 

§  189.  The  angle  which  the  radius  vector  of  the  earth  at  any  time 
would  make  with  the  same  line,  and  estimated  from  the  same  point,  had 
the  earth  moved  from  perihelion  with  its  mean  motion,  and  retained  this 
motion  unaltered,  is  called  the  mean  anomaly.  • 

§  190.  The  relation  which  connects  the  mean  with  the  true  anomaly 
'is,  Appendix  No.  V.?  equation  (g), 

n  =  V  —  2  e  sin  V  -f-  f  e9  sin  2  F  —  &c.       .     .     .     (45) 

in  which  n  is  the  i  jean  anomaly,  V  the  true  anomaly,  and  e  the  eccen- 
tricity. 

§  191.  The  difference  between  the  mean  and  the  true  anomaly  is  called 
the  equation  of  the  centre.  Denoting  the  equation  of  the  centre  by  E,  we 
have,  equation  (45), 

E  =  n  —  V  =  —  2  e  sin  V  +  f  e2  sin  2  V  —  &c.       .     .     (46) 


§  192.  Let  'S,  S.  Sa  Sw  represent  the 
ecliptic  ;  Sv  Sa  the  line  of  the  equinoxes ; 
Ss  Sw  the  line  of  the  solstices ;  Sv  the 
vernal  equinox  ;  S  the  sun  ;  PEA  EaP 
the  earth's  orbit ;  P  the  perihelion ;  A 
the  aphelion. 

When  the  earth  is  at  Ev  the  sun  will 
appear  at  the  vernal  equinox  $„;  when 
at  Et,  the  sun  will  appear  at  the  summer 
solstice  S,  \  and  when  at  Ea.  the  sun  will 


THE    EARTH'S    ORBIT.  43 

appear  at  the  autumnal  equinox  Sa ;  and  when  at  Ew,  the  sun  will  appear 
at  the  winter  solstice  Sv. 

§  193.  Let  Et  be  the  place  of  the  earth,  Em  its  mean  place ;  then  will 
SvE't,  estimated  in  the  order  of  the  signs,  that  is,  in  the  direction  indi- 
cated in  the  figure,  be  the  earth's  longitude  as  seen  from  the  sun  ;  Sv  E'm, 
estimated  in  the  same  direction,  its  mean  longitude  ;  Sv  P'  the  longitude 
of  the  perihelion  ;  P'E' t  the  true  anomaly  ;  P'E'm  its  mean  anomaly,  and 
E't  S  E'm  the  equation  of  the  centre. 

§  194.  It  is  obvious  that  the  equation  of  the  centre  is  equal  to  the  dif- 
ference between  the  mean  and  true  longitudes  from  the  same  equinox. 

§  195.  The  earth's  orbit  is  known  when  its  semi-transverse  axis,  its  ec- 
centricity, and  the  longitude  of  its  perihelion  are  known,  its  plane  being 
that  of  the  ecliptic.  These  are  called  the  elements  of  figure.  The  periodic 
time,  the  mean  motion,  and  the  mean  longitude  at  some  particular  epoch, 
are  the  additional  data  from  which  result  by  computation  the  earth's  true 
motion  and  actual  place  at  any  other  epoch  before  or  after.  These  are 
called  the  elements  of  place  and  motion. 

§  196.  Make 

L  =  mean  longitude  of  the  earth  at  the  given  epoch  ; 

t  =  an  interval  of  time  before  or  after  ; 
m  =  mean  motion  ; 

a  =  true  longitude  at  time  of  observation  ; 
ap  =  longitude  of  the  perihelion  : 

then,  Appendix  No.  V.,  equation  (i), 

L  +  m  t  =  a  —  2  e  sin  (a  —  ap)  +  f  e*  sin  2  (a  —  a^)  —  &c.      (47) 

The  sun  will  have  the  greatest  apparent  diameter  when  the  earth  is  in 
perihelion,  and  least  when  in  aphelion  ;  denote  these  diameters  by  £;  and 
df  respectively,  and  the  corresponding  radii  vectors  by  rt  and  r' ;  then 
from  the  principles  of  optics, 


and 

r'-r,:r'+ ,-,::$,-*': 
whence 

r'-r,_         S-t- 


r'+r,~      -*, 

Actual  measurements  give  about 

5,  =  32',5 
£'  =  31  '.5 


44  SPHERICAL   ASTRONOMY. 

whence  e  =  —  =  0.016  nearly  ; 

from  which  it  appears  that  e  is  so  small  as  to  justify  the  omission  from 
equation  (47)  of  those  terms  in  which  its  powers  higher  than  the  first 
enter,  and  we  may  write 

L  +  m  t  =  a  —  2  e  sin  (a  —  a,f) (48) 

§  197.  From  four  observed  right  ascensions  of  the  sun,  compute,  by 
•equation  (33),  his  corresponding  true  longitudes ;  each  longitude  increased 
by  180°  will  give  the  corresponding  true  longitude  of  the  earth ;  denote 
these  by  a,,  a^  a3,  and  a4,  and  the  intervals  of  time  from  the  epoch  of  the 
mean  longitude  Z,  say  noon,  January  1st,  to  the  times  of  observation,  by 
*,,  4,  ^3,  and  t4  respectively,  then  will,  equation  (48), 

L  +  m  t{  =  a,  —  2  e  sin  (a,  —  ap),  ^j 
L  +  m  t,  =  az  —  2  e  sin  (a,  —  a,),    I 
L  +  m  t3  =  a3  —  2  e  sin  (a3  —  a,),    j      ' 
L  -f-  m  t4  =  a4  —  2  e  sin  (a4  —  a^),  J 

four  equations,  from  which  the  mean  longitude  L  at  the  epoch,  the  mean 
motion  m,  the  eccentricity  e,  and  longitude  of  the  perihelion  ap,  may  be 
found.  For  this  purpose  subtract  the  first  from  the  second, 

m  ('*  —  $,)  =  a8  —  a,  —  2  e  [sin  (a2  —  ap)  —  sin  (a,  —  a,,)] ; 

making 

ts  —  ti  =  <J,    a2  —  a,  —  a,     or  a2  =  a,  +  a, 

and  reducing  by  the  relation 

sin  (a8—  ap)  —  sin  (a,—  ap  +  a)  =  sin  (a,—  ap)  cos  a  -f-  cos  (aj  —  ap)  sin  a, 

we  find 

m&  =  a  -\-  2  e  [sin  (a,  —  ap)  (1  —  cos  a)  —  cos  (a,  —  ap)  sin  a]  (50) 
subtracting  the  first  of  equations  (49)  from  the  third  and  fourth,  making 

t,-t,=  &',  t,  -  f,  =  d"  ; 
«3  —  <*i  =  a',  «4  —  ai  =  «"  ; 

reducing  in  the  same  way,  and  replacirg  1  —  cos  a  by  its  equal,  2  sin8  ^  a, 
we  find,  including  the  equation  above, 

m&  —  a  =  2  e  [2  sin  (a,  —  ap)  sin2  i  a  —  cos  (a,—  ap)  sin  a  "J, 
»&  6f  —  a'  =  2  e  [2  sin  (a,  —  ap)  sin9  ^  a/  —  cos  (cfl—  ap)«sin  a*  ], 
»i  d"—  a"  =  2  e  [2  sin  (a,  —  ap)  sin2  ^  a"  —  cos  (a,—  ap)  sin  a"]. 


THE   EARTH'S    ORBIT.  45 

Dividing  the  first  of  these  by  the  second  and  third  successively,  making 


m6-a       -M     1 

;^nr^  •--?• 


m  &' 

«•-.     „   \ 
snary-*!  J 

and  dividing  both  numerator  and  denominator  of  the  secord  members  by 
cos  (a,  —  ap),  we  have 

2  tan  (a,  —  ap)  .  sin3  i  a  —  sin  a 

2  tan  (a,  —  a_)  .  sin'2  \  a  —  sin  a!  ' 

p/ 
_    2  tan  (a,  -  a,)  sin2  J  a   —  sin  a     ^ 

2  tan  (at  —  ap)  sin2  i  a"  —  sin  a"  ' 
from  which  we  find 

Msrna'  —  sin  a  JVsin  a"—  sin  a 

2  tan  (a,  —  a  )  =  —     .  9  .  —  -  -  ^-.  —  =  -     .       —  -  -  .--5-=  —       (53) 
Jifsm2^  a  —  silica       .A^sin2  -i  a"—  sm2  £  a      v 

in  which  the  only  unknown  quantity  is  m;  this  entering,  equations  (51), 

the  values  of  M  and  N. 

To  find  the  value  of  m,  clear  the  fractions,  transpose  to  the  first  mem- 

ber, and  make 

n  =  sin  a  sin2  J  a1  —  sin  a'  sin9  £  a, 
&  =  sin  a  sin2  i  a"—  sin  a"  sin2  ^  a', 
*  =  sin  a"  sin2  ^  a  —  sin  a  sin2  £  u'fj  J 

or  reducing  for  the  sake  of  logarithmic  computation  by  the  relation 
sin  a  =  2  sin  ^  a  .  cos  J  a, 

n  =        2  sin  |  a     sin  \  of  .  sin  \  (ar  —  a  ),   J 

&=        2  sin  i«'.  sin  J  a",  sin  £(«"-«'),   I    >,     .     (54) 

*  =  —  2  sin  i  a     sin  i  a"  .  sin  £  («"—  «  )  *  ) 

we  have 

Mn  +  Ni  +  k.M.N=  0. 

Replacing  M  and  JVby  their  values  given  in  equations  (51),  we  find 

n  (m  &  —  a)       i(m&  —  a)  k  (m  6  —  of  _      ^ 

md'-af      h  m6"—a"        (m  &'  ~'""    ~ 


whence 

(m6—a)  [(nb"  +  i  &'  +  k  &)  m-  (n  a"  -f  ia'  -f-  £a)]  =  0. 

But  m  &  —  a  cannot  be  zero,  since  e  is  not  zero. 


46  SPHERICAL    ASTRONOMY. 

Placing,  therefore,  the  second  factor  equal  to  zero,  we  find 


a" 


n  a."  +  ^  a'  +  *#  a  w,  w  / 

=-  =  •  •  •  <55) 


From  equations  (54)  we  have 

1  _   _  ^i_?^i°  2  (a"  -  a)    } 

n  sin  ^  a'  .  sin  ^  (a'  —  a)' 

k_        sin  la",  sin  |(a".-  a')      f      ..... 

n  sin  *  a  .  sin  i-  (a'  —  a)  '    J 

Now  a,  a',  and  a''  are  the  increments  of  the   true  longitude  since  the 

•  i  k 

first  observation;  these  in  equations  (56)  give  the  fractions  -  and  -; 

these  in  equation  (55)  give  the  value  of  m\  this  in  equations  (51)  and 
(52)  give  the  value  of  tan  (a{  —  ap)  and  therefore  of  ap  ;  this,  in  equation 
(50),  gives  the  value  of  e,  and  this,  together  with  m  and  ap,  in  first  of 
equations  (49),  gives  the  value  of  L. 

§  198.  The  mean  motion  in  longitude,  the  eccentricity  and  longitude 
of  the  perihelion  being  determined  at  dates  remote  from  one  another,  are 
found  to  be  very  slightly  variable.  The  present  value  of  the  eccentricity 
is  0.01678356,  the  semi-transverse  axis  of  the  earth's  orbit,  or  the  earth's 
mean  distance  from  the  sun  being  unity  ;  that  of  the  mean  motion  in  lon- 
gitude in  one  sidereal  day  is  0°.98295603  ;  the  longitude  of  the  perigee 
at  the  beginning  of  the  present  century  was  279°  30'  05  ".0,  and  the 
mean  longitude  of  the  sun  at  the  same  time  was  280°  39'  10'''.  2. 

The  longitude  of  the  perihelion  is  found  to  increase  at  a  mean  rate  of 
61  ".9,  in  a  tropical  year,  and  deducting  50".2  for  the  retrocession  of  the 
mean  equinox,  gives  to  the  perihelion  a  direct  motion  of  11  ".7  through 
space  in  the  same  time. 

§  199.  Denoting  by  y^  the  length  of  the  tropical  year  in  sidereal  days, 
we  have 


0°.98295603 


= 

*  v     ' 


MEAN  SOLAR  TIME. 


§  200.  Although  the  mode  of  reckoning  time  by  the  motion  of  the  ver- 
nal equinox  affords  great  facilities  in  practical  astronomy,  it  is  of  little  or 
no  use  in  the  ordinary  operations  of  common  life.  Business  and  social  in- 


MEAN    SOLAR    TIME.  47 

tercourse  are  mostly  regulated  by  the  alternations  of  daylight  and  darkness, 
and  the  sun  is  the  natural  object  of  reference  in  all  divisions  of  time  for  so- 
ciety in  general. 

§  201.  Time  measured  by  the  hour  angle  of  the  sun  is  apparent  solar 
time. 

§  202.  The  epoch  of  the  sun's  being  on  the  meridian  of  a  place,  is 
called  apparent  noon  of  that  place. 

§  203.  The  interval  of  time  between  two  consecutive  passages  of  the 
sun's  centre  over  the  upper  or  lower  meridian  of  the  same  place,  is  called 
an  apparent  solar  day. 

The  apparent  solar  is  longer  than  the  sidereal  day,  in  consequence  of 
the  eartli's  real,  and  therefore  of  the  sun's  apparent,  motion  in  the  ecliptic 
in  an  easterly  direction.  If,  for  instance,  the  vernal  equinox  and  the  sun 
'were  to  pass  the  meridian  of  a  place  at  the  same  instant  to-day,  the  sun 
would  be  to  the  east  of  the  equinox  on' the  morrow,  and  would  cross  the 
same  meridian  after  it. 

§  204.  The  orbital  motion  of  the  earth  and,  therefore,  the  apparent  mo- 
tion of  the  sun  in  the  ecliptic  is,  Eq.  (a),  Appendix  V,  variable.  The 
unequal  arcs  which  measure  the  daily  increments  of  the  sun's  longitude 
vary  their  inclination  to  the  equinoctial  from  about  23°  28'  at  the  equi- 
noxes, to  zero  at  the  solstices ;  and  these  unequal  arcs  may  hence  be  pro- 
jected by  declination  circles  into  still  more  unequal  arcs  cf  right  ascension. 
These  latter  measure  the  excess  of  the  different  apparent  solar  over  the  si- 
dereal days ;  and  hence  the  variable  orbital  motion  of  the  earth,  and  the  in- 
clination of  the  plane  of  its  orbit  to  the  equinoctial,  conspire  to  make  the 
lengths  of  the  apparent  solar  days  unequal. 

§  205.  Timepieces  cannot  be  made  to  imitate  this  inequality,  nor  is  it 
desirable  they  should  do  so,  were  it  possible. 

Had  the  earth's  orbit  been  circular  and  in  the  plane  of  the  equinoctial, 
its  orbital  motion  would  have  been  uniform,  the  sun's  apparent  daily  in- 
crease of  right  ascension  constant,  and  the  apparent  solar  days  of  equal 
duration. 

§  206.  These  conditions  are  fulfilled  by  the  device  of  an  imaginary  sun 
conceived  to  move  uniformly  in  the  equinoctial  with  the  true  sun's  mean 
motion  in  longitude,  and  to  set  out  from  the  reduced  place  of  the  mean 
vernal  equinox  when  the  true  sun's  mean  place  leaves  the  mean  equinox. 

This  imaginary  body  is  called  the  mean  sun. 

§  207.  Time  measured  by  the  hour  angle  of  the  mean  sun  is  called 
mean  solar  time.  The  epoch  of  the  mean  sun  being  on  the  meridian  of  a 
place,  is  called  mean  noon  of  that  place. 


SPHERICAL    ASTRONOMY. 


Fig.  49. 


§  208.  The  difference  between  the  apparent  and  mean  -solar  time  is 
called  the  equation  of  time.  If  to  the  mean  time  the  equation  of  time  be 
applied  with  its  proper  sign,  the  apparent  time  will  result ;  if  the  equation 
of  time  be  applied  with  its  proper  sign 
to  the  apparent  time,  the  mean  time 
will  result. 

The  equation  of  time  is  employed  to 
pass  from  mean  to  apparent,  or  from 
apparent  to  mean  time. 

§  209.  Thus,  let  P  M  be  an  arc  of 
the  meridian,  VM  of  the  equinoctial, 
VE  of  the  eclipuc ;  P  the  pole  of  the 
equinoctial ;  V  the  true,  Vm  the  mean, 
and  Fr  the  reduced  place  of  the  mean 

equinox ;  S  the  true  and  Sm  the  mean  sun  ;  then  will  MP  S  be  apparent, 
and  MP  Sm  mean  solar  time ;  VSa  the  right  ascension  of  the  real  sun : 
V  Vr  the  equation  of  the  equinoxes  in  right  ascension. 

Make 

e  —  Sa  Sm  =  the  equation  of  time  ; 
a  =  VSa    =  the  right  ascension  of  true  sun  ; 
I  =  VTSm  =  the  mean  longitude  of  the  sun ; 
q  •=.  Wr    =  the  equation  of  the  equinoxes  in  right  ascension  j 

then  from  the  figure,  we  have 

e  =  a-(l  +  q) (58) 

that  is,  the  equation  of  time  is  equal  to-the  sun's  true  right  ascension  di- 
minished by  the  sun's  mean  longitude,  corrected  for  the  equation  of  the 
equinoxes  in  right  ascension. 

§  210.  When  the  sun's  true  right  ascension  exceeds  the  corrected  mean 
longitude,  the  equation  of  time  must  be  added  to  apparent  time  to  obtain 
mean  time,  and  vice  versa.  The  equation  of  time  is  zero  four  times  a  year, 
viz.,  on  15th  April,  14th  June,  31st  August,  and  24th  December. 

§  211.  The  mean  sun  and  mean  equinox  when  together  must  pnss  some 
meridian  at  the  same  instant.  .When  the  same  meridian  returns  to  the 
mean  equinox  on  the  following  day,  the  mean  sun  will  be  to  the  east  by  a 
distance  equal  to  that  which  measures  its  motion  in  one  sidereal  day ;  and 
the  mean  solar  day  will  exceed  the  sidereal  day  by  the  interval  of  sidereal 
time  required  for  the  meridian  to  overtake. the  mean  sun  after  it  passes  the 
mean  equinox. 

Denote  this  excess  by  £,  expressed  in  days ;  and  the  motion  of  the  mean 


MEAN    SOLAR    TIME.  49 

sun  in  one  sidereal  day,  equal  to  the  earth's  mean  orbital  motion  in  the 
same  time,  by  m.  Then  will  m  t  be  the  motion  of  the  mean  sun  in  the 
time  t,  and  its  right  ascension  from  the  mean  equinox  at  the  instant  the 
meridian  overtakes  it  will  be  m  +  m  t.  But  this  is  the  hour  angle  of  the 
mean  equinox,  or  the  sidereal  time  t,  reduced  to  degrees ;  whence 

m  +  mt=  360°  X  t', 
or 

t=         m        • 
360°- m' 

and  for  the  length  of  the  mean  solar  day,  expressed  in  sidereal  time, 

^  m 

+  360°  —  m' 

or  replacing  m  by  its  value  0°.98295603,  §  198,  and  denoting  the  length 
of  the  mean  solar  day  by  Dm,  expressed  in  terms  of  the  sidereal  day  D,,  as 
unity,  we  have 

Dm=  1.00273791  D (59) 

and 


Whence  to  convert  intervals  of  mean  solar  into  intervals  of  sidereal,  or  in- 
tervals of  sidereal  into  intervals  of  mean  solar  time,  we  have  these  rules, 
viz. : 

Sidereal  interval  =  1.00273791  X  Solar  interval, 

Solar  interval  =  0.99726957  X  Sidereal  interval. 

§  212.  Applying  this  second  rule  to  the  length  of  the  tropical  year  ex- 
pressed in  sidereal  days,  we  have,  Eq.  (59), 

Solar  interval  =  0.99726957  X  366.242  =  3&5.2422414  ; 

or  reducing  the  fraction  to  hours,  minutes, 'and  seconds,  and  denoting  the 
length  of  the  tropical  year,  expressed  in  mean  solar  time,  by  ytm ,  we  have 

ytm  =  365d  5h  48^  48s      .......     (60) 

§  213.  Denote  by  yam  the  length  of  the  anomalistic  year  expressed  in 
mean  solar  time;  then,  §§  157  and  198, 

360°  -  50".2  :  360°  -f-  11".  7  :  :  365d  5h  48"  48*  :  yam; 
whence 

y««=  365d  6h  13m.3 (61) 

§  214.  The  interval  of  time  required  for  the  earth  to  perfcrm  one  entire 
circuit  about  the  sun  in  space  is  called  a  sidereal  year. 


50  SPHERICAL  ASTRONOMY. 

Denote  by  ytm  the  length  of  the  sidereal  year  in  mean  solar  time,  then 

360°  —  50".2  :  360°  :  :  365d  5"  48m  48s  :  y.m; 

whence 

ym  =  365d  6h  9m  98.6 (62) 

ABERRATION. 

§  215.  The  earth's  orbital  motion,  combined  with  the  motion  of  light, 
produces  an  apparent  displacement  of  all  the  heavenly  bodies  in  the  di- 
rection of  the  point  of  the  celestial  sphere  towards  which  the  earth  is,  at 
the  instant,  moving.  This  displacement  is  called  aberration. 

Thus,  let  S  be  the  place  of  a  heavenly  Fig.  50. 

body,  E  that  of  the  earth  moving  from  M 
towards  N  along  an  arc  of  its  orbit. 
From  E  take  any  distance  E  E' ;  join  S 
and  .£",  and  lay  off  upon  E' S  the  distance 
E'  C,  which  bears  to  EE',  the  ratio  of 
the  velocity  of  light  to  that  of  the  specta- 
tor, and  suppose  C  connected  like  himself 
with  the  earth. 

Now  a  wave  of  light  from  S,  another 
which  originates  at  (7,  when  that  from  S  E      E' 

passes  this  point,  and  the  spectator's  eye 

starting  from  E  at  the  same  instant,  will  all  meet  at  E\  and  as  bodies  al- 
ways appear  in  the  direction  ot  the  normal  to  the  wave  front,  the  point  C 
and  the  body  S  will  be  seen  in  the  direction  E' S.  But  (7,  having  a  ve- 
locity equal  and  parallel  to  the  spectator,  will  have  passed  on  to  C",  at  the 
extremity  of  a  line  through  (7  equal  and  parallel  to  E  E' ;  so  that  when 
C  appears  in  the  direction  of  the  body  S,  it  will,  in  fact,  be  in  advance  of 
it  by  the  angle  S  E'  C'. 

Let  C  be  the  optical  centre  of  the  object-glass  of  a  telescope,  attached 
to  the  face  of  a  graduated  circle,  moving  in  the  plane  of  the  body  and  the 
tangent  line  to  the  terrestrial  orbit  at  the  earth's  place,  and  E'  the  inter- 
section of  the  cross  wires  at  the  solar  focus ;  then,  when  the  image  of  the 
body  appears  at  the  latter  point,  the  line  of  collimation  will  be  in  advance 
of  the  body  itself,  and  its  instrumental  bearing  will  be  in  error  by  the 
angle  S  E'  C',  and  must  be  corrected  by  the  same  angle  to  get  the  true 
bearing. 

§  216.  But  had  5  been  a  terrestrial  object,  by  the  t'nie  its  light  frcto 


ABERRATION.  51 

the  position  S  had  reached  (7,  the  body  itself  would  have  been  at  $",  the 
intersection  of  E  C  produced  and  S  S'  drawn  parallel  to  E  E' ;  and  at 
the  instant  of  its  light  reaching  E'  the  body  would  have  been  at  *S",  the 
intersection  of  S  S''  produced  and  the  line  of  collimation.  Geodetical 
observations  are,  therefore,  unaffected  by  aberration,  while  astronomical 
observations  are,  in  general,  affected  by  it. 
217.  Make 


=  aberration ; 
a  =  S  E'  N  =  angle  the  direction  of  the  body  makes  with 

that  of  the  earth's  motion. 
V  =  velocity  of  the  earth  ; 
V  =  velocity  of  light : 

Then,  in  the  triangle  G,E'  E, 

V  :  V  :  :  sin  (a  —  r)  :  sin  r, 
whence 

y 
sin  r  =  — .  sin  (a  —  r) (64) 

If  p,  denote  the  mean  radius  of  the  earth's  orbit,  then  will 

2*>      t 
365d.25636  ' 

and  it  will  be  shown  hereafter  that  light  requires  16m  26'  to  pass  over  the 
distance  2  p,,  and  therefore 


whence 

r_*Hi*xir*r 

V  365d.25636 

from  which,  and  equation  (64),  it  is  apparent  that  r  is  very  small,  and 
may  be  neglected  in  comparison  with  a ;  we  may  therefore  write 


0.00000815  sin  a, 


206264".8 
in  which  206264.8  is  the  number  of  seconds  in  radius ;  whence 

r"=  0.00009815  X  206264".8  sin  a, 
or 

r"=  20".246  sin  a (66) 


52  SPHERICAL    ASTRONOMY. 

§  218.  Let  A  B  be  the  intersection  of  the  celestial  Fig.  51. 

sphere  by  a  plane  through  the  body  and  the  direction 
of  the  earth's  motion,  A  C  that  of  a  plane  through 
the  observer  and  star,  and  perpendicular  to  the  plane 
of  the  ecliptic,  and  B  C  an  arc  of  the  ecliptic  ;  then 
will  B  be  the  point  in  which  the  tangent  to  the  earth's 
orbit  at  the  place  of  the  earth  pierces  the  celestial  sphere,  A  will  be  the 
projection  of  the  body  upon  the  celestial  sphere,  and  A  B  =  a ;  and  if 
A  C  =  X  and  C  A  B  =  9,  we  have 

cos  9  =  tan  X  cot  a, 
and 

cos2  9  =  tan*  X .  cot2  a, 
whence 


sm2  <p 
and  solving  with  respect  to  sin  a, 


tan  X  tan  X 


Vl  +  tan2  X  —  sin2  9        i/sec2  X  —  sin*  9 
and  therefore, 

sin  X 

sin  a  =  — ; 

1/1  —  cos2  X,  sin2  9 

and  this  in  equation  (65)  gives 

20".246  sin  X 
r  —  — • (66) 

1/1  —  cos2  X.  sin2  9 

which  is  the  polar  equation  of  an  ellipse,  the  pole  being  at  the  centre. 
So  that,  if  the  image  of  a  fixed  star  were  kept  constantly  on  the  cross  wires 
of  a  telescope  during  one  entire  revolution  of  the  earth  in  its  orbit,  the 
line  of  collimation  would  trace  upon  the  celestial  sphere  an  ellipse  of  which 
the  star  would  occupy  the  centre;  the  semi-transverse  axis  would  be 
20".246  and  the  eccentricity  cos  X. 

If  the  star  were  in  the  plane  of  the  ecliptic,  then  would  X  =  0, 
cos  X  =  1,  and  the  orbit  would  become  a  right  line  equal  in  length  to 
40".492.  If  the  star  were  at  either  pole  of  the  ecliptic,  then  would 
X  =  90,  cos  X  =  0,  and  the  orbit  would  be  a  circle.  Between  these  limits 
the  eccentricity  will  vary  from  1  to  0. 

The  coefficient  20".246  is  called  the  constant  of  aberration. 

§  219.  Since  the  aberration  is  in  the  arc  AB,  its  projection  on  A  C 
will  be  the  aberration  in  latitude.  Denoting  the  latter  by  r7,  we  have 


HELIOCENTRIC    PARALLAX. 

20".246  .  sin  X  .  cos  <p 


V  1  —  cos2  X.sin2  9 

which  is  obviously  the  greatest  when  <p  =  0°  or  180°,  in  which  case  the 
earth  will  be  moving  parallel  to  the  circle  of  latitude  of  the  body,  and  the 
aberration   in  latitude  will  be   equal  to  20".246  •  sin  X,  which  is  the 
semi-conjugate  axis  of  the  ellipse. 
The  aberration  in  longitude  denoted  by  r,  will  give 


20".246  .  tan  X.  sin  <p 
i/1  —  cos2X.  sin2  < 


(68) 


which  is  the  greatest  when  <p  =  90°  or  270°,  in  which  case  the  earth  will 
be  in  the  act  of  passing  the  body's  circle  of  latitude,  and  the  corresponding 
uberration  will  be  20".246,  the  semi-transverse  axis  of  the  ellipse. 

§  220.  Equations  (66),  (67),  and  (68)  are  applicable  to  a  body  which 
1ms  no  proper  motion  of  its  own.  In  case  the  body  has  a  motion,  thin 
must  be  allowed  for  in  clearing  its  instrumental  bearing  of  aberration,  and 
the  mode  of  doing  this  will  be  indicated  under  the  head  of  planets. 

§  221.  In  the  case  of  the  sun,  which  may  be  regarded  as  fixed,  9  is 
always  90°,  sin  <p  =  .1,  and  replacing  1  —  cos2X  by  sin2  X,  equation  (66), 
reduces  to 

r=20".246; 

that  is,  the  sun  will  always  appear  behind  his  true  place  by  the  constant 
of  aberration. 

§  222.  In  conclusion,  it  is  proper  to  remark  that  F,  the  velocity  of  the 
earth  in  its  orbit,  which  is  assumed  to  be  constant,  is  not  strictly  so,  but 
the  variation  is  so  small  as  not  sensibly  to  affect  the  foregoing  results. 
The  actual  velocity  varies  inversely  as  the  length  of  the  perpendicular 
drawn  from  the  sun  to  the  line  which  is  tangent  to  the  earth's  orbit  at  the 
earth's  place  (Analyt.  Mechanics,  §  193).  But  the  eccentricity  of  the  orbit 
being  very  small,  gives  but  little  variation  in  this  perpendicular. 


HELIOCENTRIC  PARALLAX. 


§  223.  The  place  in  which  a  body  would  appear  if  viewed  from  the 
centre  of  the  sun,  is  called  its  Heliocentric  place. 

§  224.  The  arc  of  a  great  circle  of  the  celestial  sphere  drawn  from  the 
heliocentric  to  the  geocentric  place  of  a  body,  is  called  its  Heliocentric 
parallax ;  and  is  obviously  the  path  a  body  would  appear  to  describe  to 


54: 


SPHERICAL    ASTRONOMY. 


an  observer  were  he  to  pass  from  one  extremity  to 
the  other  of  the  earth's  radius  vector. 

Thus,  let  S  be  the  sun,  E  the  earth,  B  the  body, 
and  MN  the  intersection  of  the  celestial  sphere 
by  a  plane  through  the  body  and  radius  vector 
SE-,  then  will  B'  be  the  heliocentric,  and  B" 
the  geocentric  place  of  the  body;  and  B" B'  its 
heliocentric  parallax.  The  heliocentric  parallax 
measures  the  angle  B"  BB'  =  SBE  =  the  angle 
at  the  body  subtended  by  the  radius  vector  of  the 
earth. 

§  225.  Make 

D  =  S  B     =  distance  of  body  from  sun ; 

R  =  SE     =  earth's  radius  vector ; 

r/  =  SBE  =  heliocentric  parallax; 

a,  =  SEB  =  angular  distance  between  sun  and  body ; 

then  in  the  triangle  SEB, 

D  :  R  :  :  sin  ay  :  sin  r/, 


whence 


R 

sin  rt  =  —  .  sin 


(69) 


When  ay  =  90°,  then  will  rt  be  the  greatest  possible.  This  maximum 
heliocentric  parallax,  is  called  the  annual  parallax  ;  which  denote  by  *, 
and  we  have 


and  if  *  be  very  small, 


sin  if  = 


it  is  expressed  in  seconds,  and  w  denotes  the  number  of  seconds  in  radius. 
From  this  we  obtain 

D  =  R.^ (VO) 

This  gives  the  distance  of  the  body  from  the  sun  in  terms  of  its  annua1 
parallax  and  the  earth's  radius  vector. 

§  226.  Substituting  the  value  of  D  in  Eq.  (69),  and  making 


we  have 


sm  r,  =  -', 

w 


r.  =  if  .  sm  a. 


(71) 


HELIOCENTRIC    PARALLAX.  55 

§  227.  Let  S  be  the  sun's  place  in  the  eclip-  Fte-  "8. 

tic,  B  the  place  of  the  body,  BA  the  arc  of  a 
circle  of  latitude,  S  A  an  arc  of  the  ecliptic, 
and  E  the  earth.  The  side  SB  will  measure 
the  angle  ay ;  and  denoting  the  side  AB  by  X, 
and  the  angle  SB  A  by  9,,  we  have 

cos  <p;  =  tan  X  .  cot  a; ; 

and  by  a  transformation,  the  same  as  in  §  218, 

sin  X 


V  1  —  cos2  X.  sin2 
which  in  Eq.  (71)  gives 


*'8inX  (72) 


V  1  —  cos2  X  .  sin2  9, 

This  is  the  polar  equation  of  an  ellipse  having  the  pole  at  the  centre ; 
and  it  shows  that  the  parallactic  path  of  a  body's  geocentric  place,  due  to 
the  earth's  orbital  motion  alone,  is  an  ellipse  of  which  the  centre  is  the 
body's  heliocentric  place. 

The  semi-transverse  axis  and  eccentricity  are  respectively  if  and  cos  X. 
If  the  body  be  in  the  pole  of  the  ecliptic,  then  will  X  =  90,  cos  X  =  0, 
and  the  ellipse  becomes  a  circle ;  if  in  the  ecliptic,  then  will  X  =  0,  cos  X 
=  1,  and  the  ellipse  becomes  a  right  line  whose  length  is  2  #. 

§  228.  Heliocentric  parallax  throws  a  body  from  its  heliocentric  place 
towards  the  geocentric  place  of  the  sun  or  towards  that  point  in  which  the 
earth's  radius  vector,  produced  beyond  the  sun,  pierces  the  celestial  sphere. 
Aberration  throws  it  towards  the  point  in  which  the  tangent  line  to  the 
earth's  orbit,  at  the  place  of  the  earth,  pierces  the  same  surface.  Both 
points  are  in  the  ecliptic,  and  if  we  neglect  the  eccentricity  of  the  earth's 
orbit,  which  we  may  do  without  sensible  error  when  the  heliocentric  par- 
allaxes are  employed,  these  points  are  90°  apart.  When,  therefore,  9  =  0° 
in  Eq.  (66),  then  will  9,  =  90°  in  Eq.  (72),  and  vice  versa  ;  and  the  least 
possible  heliocentric  parallax  will  occur  at  the  time  of  the  greatest  aberra- 
tion, and  the  least  aberration  at  the  time  of  the  annual  parallax. 

§  229.  When  the  longitudes  of  the  sun  and  body  differ  by  90°  or  270°, 
the  heliocentric  parallax  will  become  the  annual;  and  if  the  longitudes  and 
latitudes  of  the  body  be  taken  at  these  times  and  cleared  from  the  effects 
of  aberration  and  nutation,  there  will  result  the  longitudes  and  latitudes  of 
two  points  separated  by  2  *  or  double  the  annual  parallax.  The  value  of 


56  SPHERICAL   ASTRONOMY. 

<  then  becomes  known  by  a  sir  :ple  proposition  in  spherical  geometry,  and 
substituted  in  Eq.  (70)  gives  the  body's  distance  from  the  sun. 

§  230.  The  geocentric  co-ordinates  of  a  body  corrected  for  heliocentric 
parallax,  become  the  heliocentric  co-ordinates,  that  is,  the  co-ordinates  as 
they  would  appear  if  viewed  from  the  centre  of  the  sun. 

THE  SEASONS. 

§  23 1.  The  sun  is  the  great  fountain  of  those  ethereal  undulations  which, 
acting  upon  the  material  of  the  earth's  crust,  give  to  the  latter  its  surface 
heat;  and  the  temperature  of  a  place  depends  upon  its  exposure  to  their 
calorific  action.  While  the  sun  is  above  the  horizon,  the  place  is  receiving 
heat,  and  while  below,  parting  with  it ;  and  in  such  proportion  that  the 
whole  quantity  gained  and  lost  balance  each  other,  since  every  location 
has  nearly  a  constant  average  of  annual  mean  temperature,  as  indicated  by 
the  thermometer. 

§  232.  Whenever  the  sun  is  above  the  horizon  more  and  beneath  less 
than  twelve  hours,  the  general  temperature  of  the  place  will  be  above  the 
average,  and  the  converse. 

§  233.  A  portion  of  the  wave  having  a  front  surface  equal  to  unity  can 
generate  but  a  limited  quantity  of  heat,  and,  all  other  things  being  equal, 
the  temperature  at  any  one  location  will  be  inversely  proportional  to  the 
extent  of  the  earth's  surface  upon  which  this  unit  is  made  to  act.  If  the 
wave  front  be  parallel  to  the  earth's  surface  the  temperature  will  be  great- 
est, for  then  the  action  is  confined  to  the  narrowest  limits ;  if  very  oblique, 
the  temperature  will  be  low  because  the  action  is  diffused  over  a  larger 


§  234.  Let  A  B  be  the  section,  by  a  vertical  plane  through  the  sun's  cen- 
tre, of  a  portion  of  the.  wave  front,  the  surface  of  this  portion  being-unity,  say 
ten  square  miles ;  and  A  C  the  projec- 
tion of  the  same  on  the  earth's  sur- 
face by  normals  to  the  wave  front, 
called  rays. 

The  sections  are  sensibly  rectilinear 
within  the  limits  assumed,  and  the 

rays  being  normal  to  the  wave  front,   >u 

*          t  °  _ 

make  with  the  line  of  the  zenith  and 

nadir  to  the  earth's  surface,  an  angle  equal  to  BAC,  equal  to  the  sun's 

zenith  distance,  which  being  denoted  by  z,  we  have  , 

A  C  =  A  B  .  sec  z. 


THE    SEASONS.  57 

Denote  by  /'  the  temperature  when  the  wave  and  earth  surfaces  are  par- 
allel, and  by  /  when  they  are  oblique  ;  then 

AB  .  secz  :  AE  \\  I'  \  /; 
whence 

!=  —  =  !'  .cos  z; 

sec  z 

and  if  It  denote  the  temperature  which  would  result  at  the  unit's  distance 
from  the  sun,  and  r  the  radius  vector  of  the  earth,  we  have  from  the  law 
of  diffusion,  depending  upon  distance, 


whence 

/=--.  cos  z  ........     (73) 

§  235.  Resuming  Eq.  (  6  ),  and  making  p  =  90°  —  o?,  in  which  d  de- 
notes the  sun's  declination,  we  have 

cos  z  =  sin  I  .  sin  d  +  cos  /  .  cos  d  .  cos  P     .     .     .     (74) 
which,  in  Eq.  (73),  gives 

I—  ~*  -  [sm  ^  •  sin  c?  +  cos  /  .  cos  d  .  cos  P]       .     .     (75) 

This  result  is  wholly  independent  of  terrestrial  longitude,  and  is  only  de- 
pendent on  the  latitude  of  the  place,  the  sun's  declination,  and  the  place 
of  the  earth  in  its  orbit.  All  places  upon  the  same  parallel  are  equally 
exposed,  therefore,  to  the  solar  influence,  and  whatever  differences  of  mean 
temperature  and  of  climate  they  may  exhibit  are  due  to  local  causes,  such 
as  the  vicinity  of  mountains,  extended  plains,  forests,  deserts,  or  large 
bodies  of  water,  upon  all  of  which  the  sun  is  known  to  produce  great  va- 
riety of  thermal  effects. 

§  236.  Making  2  =  90°,  in  Eq.  (74),  we  have 

cos  P  =  —  tan  /  .  tan  d       ......     (76) 

and  making  P  =  0,  in  Eq.  (75),  we  have 

I=^cos(l-d)      .......     (77) 

Eq.  (76)  gives  the  value  of  the  semi-upper  diurnal  arc,  or  the  time  the  sun 
is  above  the  horizon,  or  the  duration  of  calorific  action  ;  and  Eq.  (77)  thr< 
intensity  of  the  solar  influence  when  greatest. 


58  SPHERICAL   ASTROI  OMY. 

§  237.  In  the  course  of  the  tropical  year  the  declination  varies  nearly 
47°,  the  sun  beiug  at  one  time  about  23°.5  north,  and  at  another  about 
the  same  distance  south  of  the  equator. 

As  long  as  the  latitude  and  declination  are  of  the  same  name,  that  is, 
both  north  or  both  south,  the  sun  will,  Eq.  (76),  be  longer  than  twelve 
hours  above  the  horizon,  and  the  place  will  receive  more  heat  than  it  loses. 
And  in  proportion  as  the  latitude  and  declination  approach  to  equality, 
the  intensity  of  the  solar  action  will,  Eq.  (77),  approach  its  maximum. 
This  periodical  variation  in  the  daily  average  temperature  of  a  place, 
caused  by  a  change  of  the  sun's  declination,  gives  rise  to  the  phenomena 
of  the  seasons. 

§  238.  The  interval  of  time  during  which  the  daily  increment  of  tem- 
perature of  a  place  is  increasing  is  called  its  spring ;  that  during  which 
this  increment  is  decreasing  is  called  its  summer  ;  that  during  which  the 
daily  decrement  is  increasing  is  called  its  autumn  or  fall ;  and  that  during 
which  this  decrement  is  decreasing  is  called  its  winter. 

§  239.  Within  the  tropics  C '  C'  and 
D  D',  and  especially  about  the  equator 
Q  Q',  the  temperature  is,  Eqs.  (76)  and 
(77),  nearly  uniform,  and  always  high. 
On  this  account  the  terrestrial  belt 
bounded  by  the  tropics  is  called  the 
torrid  zone. 

Between  the  tropics  and  polar  cir- 
cles A  A'  and  BE'  the  average  daily 
temperature  is  much  less  uniform  and 
always  lower  than  in  the  torrid  zone. 
The  belts  bounded  by  the  tropics  and 
polar  circles  are  called  temperate  zones. 

Between  the  poles  P  and  P'  and  polar  circles,  the  variation  of  the  av- 
erage daily  temperature  is  the  greatest  possible  and  the  temperature  itsell 
least.  The  portions  of  the  earth's  surface  about  the  poles  and  bounded  by 
the  polar  circles  are  called  frigid  zones. 

§  240.  Places  within  the  torrid  zone  may  be  said  to  have  two  of  each 
of  the  seasons  during  a  tropical  year,  and  all  places  in  the  temperate  and 
frigid  zones  but  one. 

For  all  places  in  the  north  temperate  and  frigid  zones,  spring  begins 
when  the  sun  is  on  the  equator  and  passing  from  south  to  north,  or  on  the 
20th  March ;  summer,  when  the  sun  reaches  the  tropic  of  Cancer,  or  on 
the  21st  June;  autumn,  when  the  sun  returns  to  the  equator  in  passing  to 


TRADE    WINDS.  59 

the  south,  or  22d  feeptember;  and  winter,  when  the  sun  reaches  the  tropic 
of  Capricorn,  or  21st  December.  For  all  places  in  the  south  temperate 
and  frigid  zones  the  names  of  the  seasons  will  be  reversed — spring  becomes 
autumn,  and  summer  winter. 

§  241.  The  elliptic  form  of  the  earth's  orbit  causes  the  radius  vector, 
and  therefore,  Eq.  (77),  the  intensity  of  the  solar  heat,  to  vary.  But  the 
angular  velocity  of  the  earth  about  the  sun  also  varies,  and  according  to 
the  same  law,  viz. :  that  of  the  inverse  square  of  the  earth's  distance  from 
the  sun — Analytical  Mechanics,  Eq.  (266).  Equal  amounts  of  heat  will 
therefore  be  developed  while  the  earth  is  describing  equal  arcs  of  longitude, 
and  the  supply  will  be  the  same  during  the  description  of  any  two  seg- 
ments, equal  or  unequal,  into  which  the  entire  orbit  is  divided  by  a  line 
through  the  sun.  The  earth  is  nearer  the  sun  while  the  latter  is  south  of 
the  equinoctial,  or  from  the  latter  part  of  September  to  the  latter  part  of 
March ;  and  it  describes  the  corresponding  part  of  its  orbit  in  a  time  so 
much  shortened  as  just  to  balance  the  increase  of  thermal  intensity.  But 
for  this  law  of  compensation,  the  effect  would  be  to  increase  the  difference 
of  summer  and  winter  temperature  in  the  southern  and  to  diminish  it  in 
the  northern  hemisphere.  As  it  is,  however,  no  such  inequality  is  found 
to  subsist,  but  an  equal  and  impartial  distribution  of  heat  and  light  is  ac- 
corded to  both  hemispheres. 

§  242.  But  it  must  not  be  inferred  that  the  mean  surface  heat  is  con- 
stant throughout  the  year;  for  such  is  not  the  fact.  By  taking,  at  all  sea- 
sons, the  mean  of  the  temperatures  of  places  diametrically  opposite  to  one 
another,  Professor  Dove  finds  the  mean  temperature  of  the  whole  earth's 
surface  in  June  considerably  greater  than  that  in  December.  This  is  due 
to  the  greater  amount  of  land  in  that  hemisphere  which  has  its  summer 
solstice  in  June  ;  the  thermal  effect  of  the  sun  on  land  being  greater  than 
that  on  water. 

§  243.  The  variation  of  the  radius  vector  amounts  to  about  ^  of  its 
mean  value,  and  therefore  the  fluctuation  of  heat  intensity  to  about  j1^  of 
its  average  measure — a  circumstance  which  is  manifested  in  a  great  excess 
of  local  heat  in  the  interior  of  Australia  during  a  southern,  over  that  of  the 
deserts  of  Africa  during  a  northern  summer. 


TRADE  WINDS. 

§  244.  A  discussion  of  the  trade  winds,  the  earth's  magnetism,  and  the 
tides,  belongs,  in  strictness,  rather  to  terrestrial  physics  than  to  astronomy  ; 
but  the  accessary  connection  of  these  phenomena  with  the  earth's  diurnal 


6u 


SPHERICAL    ASTRONOMY. 


rotation  and  the  action  of  foreign  bodies  upon  the  earth,  as  w«ll  as  their 
importance  to  navigation,  make  a  sufficient  apology  for  introducing  them 
here. 

§  245.  The  surface  of  the  torrid  zone  is  most  heated ;  its  excess  of 
temperature  is  communicated  to  the  superincumbent  atmosphere;  the 
latter  is  expanded,  and  becoming  specifically  lighter,  is  pressed  upward  by 
the  Bolder  portions  on  the  north  and  south  which  move  in  and  take  its 
place.  These,  in  their  turn,  are  heated,  expanded,  and  pressed  upward, 
and  a  constantly  ascending  current  is  thus  produced  ovor  an  entire  zone, 
of  which  the  boundaries  fluctuate  with  the  varying  declination  of  the  sun 
and  the  proportion  of  land  and  water  on  the  belt  of  the  earth's  crust 
lying  immediately  under  the  sun's  diurnal  path.  The  air  thus  accumu- 
lated at  the  summit  of  the  ascending  column,  being  unsupported  on  the 
north  and  south,  flows  oft'  under  the  action  of  its  own  weight  in  either  di- 
rection towards  the  poles,  and,  after  cooling,  descends  again  to  the  earth's 
surface  in  the  higher  latitudes  of  the  temperate  zones  to  supply  the  place 
nnd  follow  the  course  of  that  which  has  passed  to  the  torrid  zone. 

§  246.  Two  atmospheric  rings,  as  it  were,  distinguished  by  peculiarities 
of  internal  circulation,  are  thus  made  to  belt  the  earth  on  either  side  of 
the  equator  in  directions  paral- 
lel or  nearly  so  to  that  great 
circle.  On  the  lower  side  of 
these  rings,  in  contact  with  the 
earth,  the  air  moves  towards 
the  base  of  the  ascending  col- 
umn, and  on  the  upper  towards 
the  poles. 

§  247.  By  the  diurnal  mo- 
tion of  the  earth,  places  on  the 
equator  have  the  greatest  velo- 
city of  rotation,  and  all  other 
places  less  in  the  proportion  of 

the  radii  of  their  respective  parallels  of  latitude.  The  portions  of  the 
ascending  column  which  flow  towards  the  poles  set  out  with  the  east- 
ward intertropical  velocity,  which  they  carry  with  them  in  part  to  the 
higher  latitudes,  where  they  descend  to  the  earth's  surface.  To  an  ob- 
server situated  in  these  latitudes,  the  air  will  have  an  apparent  east- 
wardly  motion,  approaching  to  the  excess  of  the  intertropical  velocity  over 
that  of  the  observer's  parallel.  Here  westerly  winds  prevail. 

§  248.  On  parallels  a  few  degrees  lower,  the  tendency  of  the  air  is 


TEADE    WINDS  (JJ 

towards  the  equator,  and  this  combined  with  what  remains  of  the 
apparent  easterly  component,  just  referred  to,  gives  rise  in  the  north- 
ern hemisphere  to  a  northwesterly  and  in  the  southern  to  a  southwesterly 
wind. 

§  249.  In  its  onward  course  towards  the  equator,  this  same  air  crosses 
successively  parallels  of  greater  and  greater  velocity,  and  this,  together 
with  friction  against  the  earth's  surface,  reduces  the  air's  excess  of  easterly 
motion  to  zero,  and  here  northerly  winds  prevail  in  the  northern  and 
southerly  winds  in  the  southern  hemisphere. 

§  250.  In  latitu  les  still  lower,  the  excess  of  rotation  is  in  favor  of  the 
earth's  surface,  and  the  air,  unable  to  keep  up,  now  lags  behind,  and  ap- 
parently tends  to  the  west ;  and  here,  if  the  places  be  in  the  northern 
hemisphere,  northeasterly,  and  if  in  the  southern  hemisphere  southeasterly 
winds  prevail. 

§  251.  Nearer  to  the  equator  the  radii  of  the  parallels  vary  less  rap- 
idly, and  the  velocities  of  places  on  the  same  meridian  are  more  nearly 
equal.  In  crossing  these  parallels  the  air  in  its  onward  course  finds  less 
variation  in  the  velocity  of  the  earth's  surface,  and  friction,  which  now 
urges  the  air  to  the  east,  together  with  the  easterly  pressure  below,  arising 
from  the  westerly  lagging  in  the  summit  of  the  ascending  column,  due 
to  its  decreasing  angular  motion  as  it  recedes  from  the  centre  of  rotation, 
soon  brings  the  air  and  earth  to  relative  rest.  This  occurs  within  the  base 
of  the  ascending  column  where  the  currents  of  air,  which  are  continually 
approaching  each  other  from  the  directions  of  the  poles,  meet.  This  is, 
therefore,  a  region  of  calms. 

§  252.  The  aerial  currents  thus  produced  under  the  combined  influence 
of  solar 'heat  and  the  diurnal  motion  of  the  earth,  are  called  Trade  winds  ; 
and  they  are  so  called  from  the  benefits  they  are  continually  conferring  on 
trade  dependent  upon  navigation. 

§  253.  A  voyage  from  the  United  States  to  northern  Europe  in  a 
sailing  vessel  is  on  an  average  ten  days  shorter  than  in  the  contrary  direc- 
tion. A  sailing  vessel  on  a  passage  from  northern  Europe  to  the  southern 
coast  of  the  United  States  would  proceed  to  the  Madeiras  to  take  the  east- 
erly trades,  and  returning  would  proceed  to  the  Bermudas  to  catch  west- 
erly trades. 

§  254.  Within  the  region  of  calms  the  ascending  column  of  air  car- 
ries with  it  a  large  amount  of  aqueous  vapor.  In  its  ascent  the  air  expands, 
its  temperature  is  depressed,  its  aqueous  vapor  is  first  condensed  into  clouds, 
then  into  rain,  and  thus  the  region  of  calms  is  also  a  region  of  dense 
clouds  and  copous  rains  ;  the  former  giving  to  the  earth,  as  viewed  from 


62  SPHERICAL    ASTRONOMY. 

a  distance,  the  appearance  of  being  girted  tiy  dark  broken  belts,  arranged 
in  zones  parallel  to  the  equator. 

§  255.  The  limits  of  the  trades  do  not  always  occur  in  the  same  lati- 
tudes, but  vary  with  the  season.  In  December  and  January,  when  the 
sun  is  furthest  south,  the  northern  boundary  of  the  northeast  trades  of  the 
Atlantic  is  about  20°  N.,  whilst  in  the  opposite  season,  from  June  to  Sep- 
tember, it  is  32°  N. 

§  256.  Owing  to  the  great  disparity  in  the  effects  of  solar  heat  upon 
land  and  water,  and  to  the  influence  of  mountain  ranges  and  valleys  upon 
atmospheric  currents,  the  regular  trades  only  occur,  as  a  general  rule,  at 
sea,  though  in  some  level  countries,  within  or  near  the  tropics,  constant 
easterly  winds  prevail.  This  is  remarkably  the  case  over  the  vast  plains 
drained  by  the  Amazon  and  lower  Orinoco. 

§  257.  The  trades  of  the  ocean  and  of  the  land  are  separated  by  a 
belt,  within  which  other  and  variable  winds  occur.  This  belt  lies  upon 
the  ocean,  and  extends  along  the  coasts.  When  to  the  east  of  the  trades, 
it  is  often  a  hundred  miles  wide,  but  when  to  the  west  its  width  is  much 
smaller.  The  interruption  of  the  trades,  here  referred  to,  is  due  to  the 
difference  of  temperature  of  the  air  on  sea  and  land,  which  changes  with 
the  seasons.  The  air  over  the  land  in  the  higher  latitudes  is  the  warmer 
when  the  meridian  zenith  distance  of  the  sun  is  least,  and  colder  when 
greatest.  During  the  first  period  the  wind  is  from  the  sea  to  the  land, 
and  in  the  second  from  the  land  to  the  sea,  thus  giving  rise  to  the  period- 
ical winds  called  Monsoom,  which  occur  even  within  the  limits  of  the 
trades.  A  large  island  thus  circumstanced  is  surrounded  by  a  wind  blow- 
ing from  all  quarters  at  the  same  time. 

§  258.  A  similar  difference  of  temperature,  but  which  varies  with  the 
alternations  of  day  and  night,  gives  rise  to  what  are  called  the  sea  and 
land  breezes. 

TERRESTRIAL  MAGNETISM. 

§  259.  Another  most  important  effect  from  the  solar  heat,  combined 
with  the  diurnal  motion  of  the  earth,  is  the  earth's  magnetism. 

§  260.  A  difference  of  temperature  in  different  parts  of  any  body  form- 
ing a  continuous  circuit  is  ever  accompanied  by  electrical  waves,  propa- 
gated from  the  hotter  to  the  colder  parts.  If  the  circuit  be  composed  of 
various  materials,  possessing  different  powers  of  conducting  heat,  this  differ- 
ence may  be  maintained  in  greater  degree  and  duration,  and  the  effects  of 
the  electrical  flow  rendered  more  strikingly  manifest. 


TERRESTRIAL    MAGNETISM.  ^3 

§  261.  When  the  source  of  heat  is  moved  gradually  along  the  circuit, 
the  electrical  flow  is  in  the  direction  of  this  motion,  the  colder  portions 
always  lying  in  advance  and  the  warmer  behind  the  moving  source. 

§  262.  A  compass-needle,  brought  within  the  influence  of  such  a  cir- 
cuit, will  arrange  itself  at  right  angles  to  the  direction  of  the  flow,  and 
under  the  same  circumstances  the  same  end  of  the  needle  will  always 
point  in  the  same  direction.  All  this  is  the  result  of  observation  and  ex- 
periment. 

§  263.  The  earth's  crust  is  one  vast  thermo-electrical  circuit,  and  its 
source  of  heat  is  the  sun. 

§  264.  In  the  diurnal  motion  of  the  earth,  the  different  portions  of  its 
tropical  regions  are  heated  in  succession  by  the  sun  during  the  day,  and 
cooled  by  radiation  during  the  succeeding  night.  The  hotter  portions  will 
therefore  lie  to  the  east  and  the  colder  to  the  west  of  the  sun's  place.  A 
perpetual  flow  of  electricity  is  thus  developed  and  maintained  in  and 
about  the  earth's  crust  from  east  to  west,  and  gives  rise  to  the  earth's 
magnetic  action. 

§  265.  Were  the  materials  of  the  earth  all  equally  good  electrical  con- 
ductors, and  the  sun  always  in  the  equinoctial,  the  electrical  flow  would  be 
parallel  to  that  great  circle,  and  the  compass-needle  would  always  point 
directly  north  and  south.  But  neither  of  these  conditions  obtains.  The 
materials  vary  greatly  in  conducting  power,  and  the  sun's  declination  is 
ever  changing. 

§  266.  The  disparity  of  conducting  power  directs  the  electrical  flow  in 
paths  of  double  curvature,  of  which  the  general  direction  is  parallel  to 
the  equator,  and  the  varying  declinations  of  the  sun  are  perpetually  shift- 
ing their  precise  location  and  shape  as  well  as  changing  the  intensity  of 
the  flow. 

§  267.  The  position  of  stable  equilibrium,  assumed  by  a  magnetic  nee- 
dle reduced  to  its  axis,  freely  suspended  from  its  centre  of  gravity,  and  sub- 
jected alone  to  the  directive  action  of  the  earth's  magnetism,  is  called  the 
magnetic  position  of  the  place. 

§  268.  The  intersection  by  a  vertical  plane  throifgh  the  magnetic  posi- 
tion with  the  celestial  sphere,  is  called  the  magnetic  meridian. 

§  269.  The  angle  made  by  the  magnetic  and  the  true  meridian  is 
called  the  magnetic  declination,  or  simply  declination. 

§  270.  The  inclination  of  the  magnetic  position  to  the  hrrizon  is  called 
the  magnetic  inclination  or  dip. 

§  271.  The  magnetic  position  at  the  same  place  is  continually  varying 
Tt  describes  daily  a  conical  surface,  of  which  the  place  is  the  vertex,  and 


(J4  SPHERICAL    ASTRONOMY. 

daily  mean  position  the  axis,  while  this  axis  itself  describes  a  similar  sur- 
face once  a  year  about  an  annual  mean  position. 

§  272.  The  mean  of  all  the  declinations  and  of  dips  throughout  any 
one  day  are  the  declination  and  dip  for  that  day,  and  are  called  th<» 
diurnal  declination  and  dip.  The  mean  of  all  the  diurnal  declination* 
and  dips  for  the  different  days  throughout  any  given  year,  are  the  decli- 
nation and  dip  for  that  year,  and  are  called  the  annual  declination 
and  dip. 

§  273.  The  daily  and  annual  fluctuations  here  referred  to  are  called 
periodic  changes.  The  annual  de.clination  and  dip  also  change,  and  these 
changes,  which  are  found  to  take  place  in  the  same  direction  for  a  great 
many  years,  are  called  secular  changes. 

§  274.  The  magnetic  declination  and  dip  vary,  in  general,  with  the 
locality.  The  line  connecting  those  places  where  the  declination  is  zero, 
is  called  the  line  of  no  declination  ;  and  the  line  through  the  placer,  where 
the  dip  is  zero,  is  called  the  magnetic  equator. 


Fig.  5T. 


§  275.  According  to  the  Magnetic  Atlas  of  Hansteen,  constructed  for 
1787,  the  line  of  no  declination  is  found  on  the  parallel  of  60°  north,  a 
little  to  the  west  of  Hudson's  Bay;  it  proceeds  in  a  southeasterly  direc- 
tion, through  British  America,  the  northwestern  lakes,  the  United  States, 
and  enters  the  Atlantic  Ocean  near  Chesapeake  Bay,  passes  near  the  An- 
tilles and  Cape  St.  Roque,  and  continues  on  through  the  southern  Atlantic 
till  it  cuts  the  meridian  of  Greenwich  in  south  latitude  05°.  It  reappears 
in  latitude  60°  south,  below  New  Holland,  crosses  that  island  through  its 
centre,  runs  up  through  the  Indian  Archipelago  with  a  double  sinuosity, 
and  crosses  the  equator  three  times — first  to  the  east  of  Borneo,  then  be- 
tween Sumatra  and  Borneo,  and  again  south  of  Ceylon,  from  which  it 
passes  to  the  east  through  the  Yellow  Sea.  It  then  stretches  across  the 


TLKRESTRIAL    MAGNETISM.  05 

coast  of  China,  making  a  semicircular  sweep  to  the  west  till  it  reaches 
the  parallel  of  71°  north,  when  it  descends  again  to  the  south,  and  re- 
turns northward  with  a  great  semicircular  bend,  which  terminates  in  the 
White  Sea. 

On  the  magnetic  chart  this  line  is  accompanied  through  all  its  windings- 
l>y  other  lines  upon  which  the  declination  is  5°,  10°,  15°,  &c. ;  the  latter 
becoming  more  irregular  as  they  recede  from  the  line  of  no  declination. 
The  use  of  these  lines  is  to  point  out  to  navigators  sailing  by  compass,  the- 
bearing  of  the  true  meridian  from  the  magnetic. 

§  276.  On  the  east  of  the  American  and  west  of  the  Asiatic  branch  of 
the  line  of  no  declination,  the  declination  is  west,  while  to  the  west  of  the 
American  and  east  of  the  Asiatic  branch  the  declination  is  east. 

§  277.  The  magnetic  equator  cuts  the  terrestrial  equator,  according  to 
Ilansteen,  in  four,  and  to  Morlet  in  two  points,  called  nodes,  one  of  which 
is  in  the  centre  of  Africa. 

§  278.  Beginning  at  the  African  node  the  magnetic  equator  advances 
rapidly  to  the  north,  and  quits  Africa  a  little  south  of  Cape  Guardafui,  and 
attains  its  greatest  north  latitude,  12°,  in  62°  of  east  longitude  from  Green- 
wich. Between  this  meridian  and  174°  east,  the  magnetic  is  constantly 
to  the  north  of  the  terrestrial  equator.  It  cuts  the  Indian  peninsula  a 
little  to  the  north  of  Cape  Comorin,  traverses  the  Gulf  of  Bengal,  making 
a  slight  advance  to  the  terrestrial  equator,  from  which  it  is  only  8°  distant 
a  its  entrance  into  the  Gulf  of  Siam.  It  here  turns  again  a  little  to  the 
north,  almost  touches  the  north  point  of  Borneo,  traverses  the  straits  be- 
tween the  Philippines  and  the  isle  of  Mindanao,  and  on  the  meridian  of 
Naigion  it  again  reaches  the  north  latitude  of  9°.  From  this  point  it 
traverses  the  archipelago  of  the  Caroline  Islands,  and  descends  rapidly  to 
the  terrestrial  equator,  which  it  cuts,  according  to  Morlet  in  174°,  and 
according  to  Hansteen  in  187°  east  longitude.  Its  next  point  of  contact 
with  the  equator  is  in  west  longitude  120°.  Here,  according  to  Morlet,  it 
does  not  pass  into  the  northern  hemisphere,  but  bends  again  to  the  south, 
while  Hansteen  makes  it  cross  to  the  north,  and  continue  there  for  a  dis- 
tance of  15°  of  longitude,  and  then  return  southward  and  enter  the  south- 
ern hemisphere  in  longitude  108°  west,  or  23°  from  the  west  coast  of 
America.  Between  this  point  and  its  intersection  with  the  terrestrial 
equator  in  Africa,  the  magnetic  equator  lies  wholly  in  the  southern  hemi- 
sphere, its  greatest  southern  latitude  being  about  25°. 

§  279.  The  dip  increases  as  the  needle  recedes  on  either  side  from  the 
magnetic  equator,  the  end  of  the  needle  which  was  uppermost  in  the 
northern  being  lowermost  in  the  southern  hemisphere. 

5 


C6 


SPHERICAL    ASTRONOMY. 


§  280.  The  points  at  which  the  magnetic  needle  is  vertical  are  called 
the  magnetic  poles.  Of  these  there  are  four,  two  in  each  hemisphere, 
their  positions  being  indicated  on  the  magnetic  charts. 

§  281.  On  the  magnetic  charts,  the  magnetic  equator  is  accom 
pauied  by  curves  of  equal  jp  as  in  the  case  of  the  lines  of  equal  decli 
nation. 

§  282.  The  line  of  no  declination  and  the  nodes  of  the  magnetic  equa- 
tor are  found  to  have  a  slow  westerly  motion,  thus  causing  the  differ- 
ent lines  of  equal  declination  and  dip  to  pass  successively  through  the 
same  place,  and  illustrating  the  utter  worthlessness  of  all  maps  constructed 
from  compass  bearings  unless  the  diurnal  declinations  of  the  needle  are 
carefully  ascertained  and  recorded  thereon. 

§  283.  The  intensity  of  the  earth's  magnetic  action  increases  with  the 
proximity  of  the  electrical  paths  to  the  needle  and  with  the  difference  o( 
temperature  in  their  different  parts  ;  and  from  changes  in  these,  produced 
by  the  varying  zenith  distance  of  the  sun  during  the  day,  and  of  his  me- 
ridian zenith  distance  throughout  the  year,  arise  the  daily  and  annual 
mutations  of  declination  and  dip ;  while  to  changes  of  the  earth's  crust, 
produced  by  geological  causes,  and  increased  cultivation  of  the  soil  from 
the  spread  of  civilization,  are  to  be  attributed  the  secular  variations  of  the 
same  elements. 


TIDES.     • 

§  284.  Those  periodical  elevations  and  de- 
pressions of  the  ocean  by  which  its  waters  are 
made  to  flow  back  and  forth  through  the 
estuaries  that  indent  our  coasts,  are  called 
Tides. 

§  285.  Perpetual  change  in  the  weight  of 
the  waters  of  the  ocean,  due  to  the  attraction 
of  the  sun  and  noon  upon  the  earth,  and  the 
diurnal  rotation  of  the  latter  about  its  axis, 
cause  and  mairtvn  the  tides. 

§  28G.  Let  AGED  be  a  great  circle  of 
the  earth,  in  a  plane  through  the  sun's  centre 
at  S.  Draw  S  E  through  the  earth's  centre 
at  E,  and  CD  through  the  same  point,  and  at 
right  angles  to  S  E.  Assume  any  unit  of 
mass  as  that  at  #;  join  G  and  S,  and  make 


TIDES.  (J7 

d  =  S  E     =  distance  of  sun  from  the  earth  ; 
p  =  E  G      ~  radius  of  the  earth  ; 
z  =  S  G      =  distance  of  G  from  sun  ; 
<p  =.  A  E  G  =  angular  distance  of  G  from  sun  ; 
$  =  G  S  E  =  angle  at  sun  subtended  by  radius  p  ; 
m  =  mass  of  sun  ; 
k  —  the  attraction  of  unit  of  mass  at  unit's  distance. 

Then,  since  the  attraction  on  unit  of  mass  is  proportional  to  the  attract- 
ing mass  directly,  and  the  square  of  the  distance  inversely,  the  sun's  action 
on  G  will  be 

km 

~^; 
or  because 

s?  —  d8  +  p*  —  2  d  p  cos  <p, 

km 


<F  -f  p8  —  2  d  p  cos  <f 

But  each  unit  of  the  earth's  mass  is  acted  upon  by  a  centrifugal  force 
equal  and  contrary  to  the  centripetal  force  impressed  upon  the  unit  of 
mass  to  deflect  it  from  its  tangential  into  its  orbital  path.  This  latter  is, 
by  making  p  =  0,  in  the  above 

km 


applying  this  to  G  in  the  direction  G  H  parallel  to  S  E,  we  have  all 
the  action  on  G  arising  from  the  sun's  attraction. 

Resolving  these  forces  into  their  components  in  the  direction  of  the 
radius  E  G,  and  perpendicular  thereto  ;  also  making 

v  =  resultant  of  the  components  in  direction  .of  the  radius, 
r  =:       "  "  "  "         of  tangent, 

and  regarding  the  components  which  act  towards  the  centre  as  positive 
and  the  contrary  negative  ;  also  the  tangential  components  which  act  in 
the  direction  A  G  C  B  A  as  positive  and  the  contrary  negative,  we  Imve 

km  km  / 


» 

d3  d*  -f  p2  —  2  d  p  cos  9 

km   .  km  •    /         »\ 

r  =  -35-  sm  9  —  -^  -  =  -  —  j—     —  .sin(<p-M)     .          (.!'« 
d8  t^-j-p'—  2o?pcos<p 


68  SPHERICAL    ASTRONOMY. 

Developing  the  last  factor  in  equation  (  78  ),  making  cos  d  =  1,  because  of 
the  small  value  of  $,  we  have,  after  reducing, 

2  k  m  .  p                                  p                                    km 
v=. .  (cos*  $ 4- .  cos  0)  -\ .gin  4  .  sm  0 


but  from  the  triangle  E  G  S,  we  have 

p  .  sin  (<p 


sm     = 


a 

yr  neglecting  6  in  the  second  member 

p  .  sin  <p 

sm  6  =  -     ,   r. 
a 

Substituting  this  and  omitting  all  the  terms  into   which  £  enter's  as  a 
factor,  which  we  may  do  without  materially  altering  the  value  of  v,  we  find 

2  k  m  .  p  km  a 

v=  --  __P.cos>  +  _P.sin2<p  .     .     .     .       (80) 

Again,  ^mitting  6,  in  the  last  factor  of  equation  (t9),  reducing  to  a 
common  denominator  and  neglecting  the  terms  of  which  ~  is  a  factor,  we 
have,  after  replacing  cos  9  .  sin  <p  by  J  sin  2  (p, 


(81) 


§  287.  Making  9  =  0°  and  (p  =  180°  in  equation    (80),  we  have  the 
effect  on  the  waters  at  A  and  at  B  ;  and  in  both  cases 


Again,  making  <p  =  90°  and  (p  =  270°,  we  have  the  effect  on  the  wa- 
ters at  C  and  D ;  and  in  both  cases 

km  p 

~dT' 

The  values  of  v  at  A  and  B  being  negative  and  those  at  0  and  D  posi- 
tive, and  these  being  connected  by  a  law  of  continuity,  through  equation 
(80),  the  effect  of  the  sun's  attraction  is  to  increase  the  weight  of  the 
unit  of  mass,  or,  what  is  the  same  thing,  the  specific  gravities  of  all  bodies 
gradually,  in  both  directions,  from  A  to  C  and  D,  and  to  diminish  them 


TIDES. 


in  like  manner  from  <7  and  D  to  B. 
And  this  being  true  of  all  sections  of 
the  earth  through  its  centre  and  the 
sun,  the  waters  of  the  ocean  on  and 
near  the  circumference  of  a  section 
through  th3  earth's  centre,  and  perpen- 
dicular to  these,  will,  by  the  principles 
of  hydraulics,  press  up  those  about  A 
and  B  till  their  increased  height  shall 
compensate  for  their  diminished  speci- 
fic gravity,  or  till  the  weights  of  the 
balancing  columns  become  equal;  so 

that  the  ocean  surface  will  tend  to  assume,  as  its  form  of  equilibrium,  that 
of  an  oblongated  ellipsoid,  of  which  the  longer  axis  is  directed  towards 
the  sun.  The  difference  of  the  longer  and  shorter  semi-axes  of  this  ellip- 
soid is  about  23  inches. 

§  288.  If  the  earth  had  no  diurnal  rotation  about  its  axis,  this  ellipsoid 
of  equilibrium  would  be  formed,  and  all  would  be  permanent.  But  th« 
earth's  diurnal  and  orbital  motion,  together  with  the  inertia  of  water,  leave 
no  sufficient  time  for  this  spheroid  to  be  fully  formed.  Before  the  watei* 
can  take  their  level,  these  motions  carry  the  line  connecting  the  earth  and 
sun  westwardly,  and  the  place  of  the  vertex  of  the  spheroid  of  equilibrium 
in  the  same  direction,  thus  leaving  that  of  the  actual  spheroid  to  the  east 
of  the  sun,  and  forcing  the  ocean  to  be  ever  seeking  a  new  bearing.  The 
effect  is  to  produce  an  immensely  broad  and  excessively  flat  wave,  which 
follows  or  endeavors  to  follow  the  apparent  diurnal  motion  of  the  sun,  and 
completes  an  entire  circuit  of  the  earth  once  in  twenty-four  solar  hours, 
thus  producing  a  rise  and  fall  of  the  ocean  level  twice  within  this  period  on 
every  meridian. 

§  289.  The  rising  water  is  called  the  flood,  the  falling  the  ebb  tides,  and 
the  general  swell  of  the  ocean  is  called  the  primitive  tide- wave. 

§  290.  In  the  open  ocean,  where  the  water  is  deep,  and  therefore  per- 
mits the  free  transmission  of  pressure  from  one  remote  point  to  another, 
the  motion  is  one  of  oscillation  in  a  vertical  direction  principally.  But 
where  the  tide-wave  approaches  shoals,  such  as  those  along  the  coasts  and 
the  beds  of  estuaries,  which  intercept  the  free  transmission  of  pressure,  t.h« 
water  becomes  piled  up,  as  it  were,  on  the  side  of  the  open  ocean,  without 
being  able  to  press  up  any  thing  to  its  support  on  the  land  side.  It  there- 
fore flows  inland,  an<?  produces  what  are  called  derivative  flood  tides. 
After  the  apex  of  the  tide-wave  has  passed  onward,  and  low-water  sue- 


70  SPHERICAL    ASTRONOMY. 

ceeds,  the  want  of  support  is  transferred  tc  the  side  of  the  ocean,  the 
water  flows  out  to  sea,  and  forms  what  are  called  derivative  ebb  tides. 
The  lines  on  the  earth's  surface  connecting  thcee  places  at  which  high  01 
low  water,  or  any  other  corresponding  phases  of  the  tides,  occur  simulta- 
neously, are  called  cotidal  lines. 

§  291.  The  earth  and  moon  are  so  near  to  each  other,  and  so  remote 
from  the  sun,  as  to  cause  their  mutual  attractions  greatly  to  predominate 
over  the  excess  of  the  sun's  attraction  for  one  of  them  over  his  attraction 
for  the  other.  They  therefore  revolve  about  their  common  centre  of 
gravity,  and  together  move  around  the  sun.  The  attraction  of  the  moon 
for  the  earth  produces  upon  the  ocean  effects  similar  to  those  of  the  sun. 

§  292.  The  diminution  of  weight  at  A  and  B  and  increase  at  C  and 
D  vary  directly  as  the  attracting  masses,  and  inversely  as  the  cubes  of 
their  distance,  equations  (80)  and  (81),  and  the  effects  upon  the  tide- 
wave  must  be  in  the  same  proportion.  The  mass  of  the  sun  is  355000  x  88 
that  of  the  moon,  and  he  is  situated  at  400  times  the  moon's  distance. 
Whence  the  effect  of  the  moon  at  A  and  B  being 

'-p> 

lhat  of  the  sun  will  be 

2A:pra355000x88a 
~(400)3 13  * 

and  dividing  the  last  by  the  first,  we  have 
355000  X  88 


(400)3 


=  0.488 ; 


so  that  the  effect  of  the  moon  is  more  than  double  that  of  the  sun. 

§  293.  The  lunar  day  exceeds  the  solar  on  an  average  about  50  min- 
utes ;  the  lunar  tide  must  therefore  move  slower  than  the  solar  by  about 
1 2°.5  in  24  solar  hours ;  and  hence  they  must  sometimes  conspire  and 
sometimes  oppose  one  another.  The  former  occurs  when  the  angular  dis- 
tance of  the  sun  from  the  moon,  as  seen  from  the  earth,  is  0°  or  180°,  and 
(he  latter  when  this  distance  is  90°. 

This  alternate  reinforcement  and  partial  destruction  of  the  lunai  by  the 
solar  wave,  produce  what  are  called  spring  and  neap  tides ;  the  former 
being  their  sum,  the  latter  their  difference. 

§  294.  The  sun  and  moon,  by  virtue  of  the  ellipticities  of  the  terres- 
trial and  lunar  orbits,  are  alternately  nearer  to  and  further  from  the  earth 
than  their  mean  distances. 


TIDES.  71 

If  the  mean  distances  of  the  sun  and  moon  be  substituted  in  Eq.  (80), 
the  corresponding  ellipticities  of  the  solar  and  lunar  spheroids  will  be  found 
to  be  2  and  5  feet  respectively ;  so  that  the  average  spring  tide  will  be  to 
the  average  neap,  as  5  +  2  to  5  —  2,  or  as  7  to  3." 

Substituting  the  greatest  and  least  distance  of  the  sun  in  the  same 
equation,  the  resulting  tides  are  called  respectively  apoyean  and  pe^igean 
tides ;  and  representing  the  ellipticity  of  the  solar  spheroid  at  the  mean 
distance  by  20,  the  corresponding  ellipticities  become  19  and  21.  In  like 
manner  the  ellipticities  of  the  lunar  spheroid  will  be  found  to  vary  be- 
tween the'limits  43  and  59.  Hence,  the  highest  spring  tide  will  be  to  the 
lowest  neap,  as  59  +  21  is  to  43  —  21,  or  as  10  to  2,8. 

§  295.  The  sun  and  moon  act  to  form  the  apexes  of  their  respective 
tide-waves  at  different  places,  depending  upon  their  angular  distances 
apart  This  gives  rise  to  a  resultant  wave,  whose  apex  is  at  some  inter- 
mediate place,  and  the  actual  tide  day,  or  interval  between  the  occurrences 
of  two  consecutive  maxima  of  the  resultant  w#ve  at  the  same  place,  will 
vary  as  the  component  waves  approach  to  or  recede  from  one  another. 
This  variation  from  uniformity  in  the  length  of  the  tide  day  is  called  the 
priming  or  lagging  of  the  tides — the  former  indicating  an  acceleration  and 
the  latter  a  retardation  of  the  recurrence  of  high-water  at  the  same  place. 
The  priming  and  lagging  are  particularly  noticeable  about  the  time  the 
angular  distance  between  the  moon  and  sun  is  0°  or  180°,  that  is,  as  we 
shall  presently  see,  about  new  or  full  noon. 

§  296.  The  effort  of  the  attracting  body  being  to  form  the  nearest  ver- 
tex of  its  aqueous  spheroid  immediately  under  it,  the  summit  of  the  lunar 
and  solar  tide-waves  follow  the  course  of  the  moon  and  sun  to  the  north 
and  south  of  the  equator,  and  this  gives  rise  to  a  monthly  and  annual 
variation  in  the  heights  of  the  pnncipal  tides  at  a  given  place. 

§  297.  But  of  all  causes  of  difference  in  the  heights  of  tides,  local 
situation  is  the  most  influential.  In  some  places,  the  tide-wave  rushing  up 
narrow  channels  becomes  so  compressed  laterally  as  to  be  elevated  to  extra- 
)rdinary  heights.  At  Annapolis,  in  the  Bay  of  Fundy,  it  is  said  to  rise 
120  feet. 

§  298.  Were  the  waters  of  the  ocean  free  from  obstmctions  due  to 
viscosity,  friction,  narrowness  of  channels  leading  to  different  ports,  and 
the  like,  the  time  of  high-water  at  a  given  place,  would  depend  only  upon 
the  relative  positions  of  the  sun  and  moon,  and  their  meridian  passages. 
But  all  these  causes  tend  to  vary  this  time,  and  to  postpone  it  unequally  at 
different  ports.  This  deviation  of  the  time  of  actual  from  that  of  theoret- 
ical high-water  at  any  ph;ce,  is  called  the  establishment  of  the  port,  and  is 


72  SPHERICAL    ASTRONOMY. 

an  element  of  the  highest  maritime  importance.  When  ascertained  from 
observation,  it  enables  the  manner  to  know  by  simply  noticing  the  places 
of  the  sun  and  moon  with  reference  to  the  meridian,  when  ne  may  safely 
attempt  the  entrance  of  a  port  obstructed  by  shoals. 

§  299.  In  bays,  rivers,  and  sounds,  where  tides  arise  from  an  actual 
flow  of  water,  the  time  of  "  Slack  water"  or  stagnation,  must  not  be  con- 
founded with  that  of  high  and  low  water.  They  may,  indeed,  coincide, 
but  not  of  course.  A  river  current,  for  instance,  and  another  from  the  sea, 
»«...,•  neutralize  each  other's  flow,  while  both  conspire  to  elevate  the  water 
surface;  so,  also,  an  ebbing  current  may  continue  its  onward  course  after 
the  more  advanced  part  of  a  returning  flood  has  put  its  surface  on  the  rise 
by  checking  its  velocity.  The  same  of  two  currents  meeting  in  a  sound. 

§  300.  Starting  from  A  as  an  origin  (Fig.  58),  and  proceeding  in  the 
direction  of  A  C  B  D  A,  we  find  the  value  of  T,  Eq.  (81),  negative  in 
the  1st  and  3d  quadrants,  and  positive  in  the  2d  and  4th  ;  so  that  th*» 
tangential  components  of  the  solar  and  lunar  attractions  conspire  with  the 
normal  to  increase  the  height  of  the  gpeat  tide-waves  by  impressing  upon 
the  water  a  motion  of  translation  towards  their  apexes.  But  before  the 
inertia  of  the  water  will  permit  the  latter  to  acquire  much  velocity,  the 
rotary  motion  of  the  earth  reverses  the  direction  of  the  impelling  forces, 
and  the  final  effect  due  to  this  cause  is,  in  consequence,  but  small. 


TWILIGHT. 

§  301.  The  curve  along  which  a  conical  surface,  tangent  to  the  sun  and 
earth,  is  in  contact  with  the  latter  body,  is  called  the  circle  of  illumination. 
It  divides  the  dark  from  the  enlightened  portion  of  the  earth's  surface,  and 
is  ever  shifting  its  place  by  the  diurnal  motion. 

§  302.  The  base  of  the  earth's  shadow,  into  which  a  spectator  enters  at 
sunset,  and  from  which  he  emerges  at  sunrise,  is  inclosed  by  an  atmospheric 
wall-like  ring,  illuminated  by  the  direct  light  from  the  sun,  immediately 
exterior  to  that  which  just  grazes  the  earth's  surface.  The  light  is  reflected 
from  the  particles  of  this  ring  into  the  shadow,  and  gives  to  the  air  about 
its  boundary  a  secondary  and  partial  illumination  called  Twilight.  A  co- 
nical surface  through  the  summit  of  this  ring,  and  tangent  to  the  earth, 
determines,  by  its  contact  with  the  latter,  a  limit  within  which  the  twilight 
cannot  sensibly  enter,  and  twilight  will  only  continue  while  the  spectatoi 
is  carried  by  the  earth's  diurnal  motion  across  the  zone  of  which  this  line 
is  the  inner,  and  the  circle  of  illumination  the  exterior  bounda-v.  The 


TWILIGHT 


73 


belt  of  the  earth's  surface  over  which  twilight  is  visible,  is  called  the  cre- 
puscular zone. 

Thus,  let  E  0  Of  E'  be  a  section  Fi?- 60- 

of  the  earth's  surface  on  the  opposite 
side  from  the  sun  ;  TAA'  T'  of  the 
atmosphere  by  the  same  plane,  the 
height  of  the  air  being  exaggerated 
to  avoid  confusing  the  figure;  and 
S  A  and  S'  A'  two  solar  rays  tan- 
gent to  the  earth's  surface.  The 
particles  of  air  in  EA  T  and  E'A'T 
will  be  illuminated,  while  those  in 
fbe  space  EAA'E'  will  be  in  the 
shadow.  The  section  will  cut  from 
the  tangent  cone  the  elements  A  V 
and  A'  V,  which  touch  the  earth  at 
0  and  0',  respectively,  and  being 
revolved  about  the  line  connecting 

the  centres  of  the  earth  and  sun,  the  part  EA  T  will  generate  the  lumin- 
ous atmospheric  inclosure  and  the  points  E  and  0,  the  circle  of  illumina- 
tion and  interior  boundary  of  the  crepuscular  zone,  respectively. 

§  303.  To  a  spectator  within  the  crepuscular  zone  a  portion  only  of  the 
illuminating  ring  will  be  visible,  and  will  appear  as  a  bright  elliptical  seg- 
ment, with  its  chord  in  the  horizon,  its  vertex  in  the  vertical  circle  through 
the  sun,  and  its  outline  almost  lost  in  the  gradual  decay  of  light  produced 
by  the  diffusive  action  of  the  air  and  the  progressive  thinning  and  conse- 
quent diminution  in  the  number  of  reflecting  particles  towards  the  summit 
of  the  luminous  ring. 

§  304.  When  the  spectator  is  carried  obliquely  through  the  crepuscular 
zone  without  crossing  its  smaller  base,  twilight  will  last  all  night. 

§  305.  Resuming  Eq.  (74),  that  is 

cos  z  =  sin  I  sin  d  -f  cos  I  cos  d  cos  P ; 

substituting  the  latitude  of  the  place  for  £,  the  declination  of  the  sun  for  rf, 
and  the  value  of  P,  obtained  by  converting  the  observed  time  from  noon 
to  the  end  of  twilight  in  the  evening,  or  from  the  beginning  of  twilight  in 
the  morning  till  noon,  into  degrees,  the  average  value  of  a  number  of  de- 
terminations for  z  will  be  found  to  be  about  108°;  so  that  at  the  end  of 
evening  or  beginning  of  morning  twilight  the  sun  is  18°  below  the  hor'zoiL 
§  306.  From  the  above  equation  we  find 


74  SPHERICAL    ASTRONOMY. 

COS  2  —  COS  I  .  COS  d  .  COS  P 


sn     = 


sin  d 


The  angle  P  S  Z,  made  by  the  hour 
circle  P  S  and  vertical  circle  Z  S,  is 
called  the  variation  or  the  parallactic 
angle.  Denote  this  by  £,  then  from  the 
triangle  Z  P  S,  will 

( 1 )       .      .      .    sin  I  •=.  sin  d  cos  z  -f-  cos  d  sin  g  cos 

Equating  the  second  members  of  this 
and  the  equation  above,  we  have 


(2)  ....     cos  I .  cos  P  =  cos  z .  cos  d  —  sin  z  sin  d .  cos  £; 
and  if  the  sun  be  in  the  horizon,  then  will 

z  =  90°,  P  =  P',  and  (  =  ?,  and 

(3)  ....     cos  I .  cos  P'  =.  —  sin  d  .  cos  £'. 
Also,  from  the  same  triangle,  * 

(4)  ....     cos  I.  sin  P  =  sin  z  .sin  £; 
and  when  the  sun  is  in  the  horizon, 

(5)  .     .     .     .     cos  I .  sin  P'  =  sin  f . 

Multiply  (2)  by  (3),  also  (4)  by  (5),  and  add  the  products,  there  will 
result, 

cos»  I .  cos  (P—  P1)  =  —  cos  z  cos  d  sin  d  cos  f  -f  sin  z  cos  (£  —  f )  —  cos»  d  sin  a  cos  £  cos  f . 

From  (1),  we  have 

sin  /  —  sin  d  .  cos  z 


cos     = 


cos  a  .  sin 


and  for  the  sun  in  the  horizon 


COS  P  i= 


sin  / 


(82) 


(83) 


TWILIGHT. 

which  substituted  above,  give  J/RNM. 

cos2  / .  cos  (P  —  P')  =  sin  z  .  cos  (£  —  £')  —  sin8  / ; 
whence,  because 

cos  (P  -  P')  =  1  -  2  sin2  \  (P  -  P'), 
we  have 

1  —  sin  z  .  cos  (B  —  £')  ^ 


sin2  J  (P  -  P')  = 

2  cos2 

passing  to  the  arc  and  making 

_P  —  P' 

~15~' 
we  have 


which  will  give  the  time  required  for  the  sun,  or  other  heavenly  body, 
to  pass  from  the  horizon  to  a  zenith  distance  z,  or,  conversely,  from  a 
zenith  distance  z  to  the  horizon. 

Making  z  =  90°  -f  18°  =  108°,  Eq.  (84)  becomes 


which  will  give  the  duration  of  twilight  for  any  latitude  and  season  of 
the  year ;  and  for  this  purpose,  the  values  of  £  and  £'  must  be  found 
from  Eqs.  (82)  and  (83),  after  making,  in  the  former,  z  =  90°  +  18°. 

The  value  of  /,  in  Eq.  (85),  becomes  a  minimum  when  £  =  £',  and 
for  the  duration  of  the  shortest  twilight,  we  have,  after  replacing 
1  —  cos  18°  by  its  equal  2  sin2  9°, 

t  —  —  .  sin"1  (sin  9°  .  sec  /) (86) 

lo  • 

Equating  the  second  members  of  Eqs.  (82)  and  (83) 

sin  d  -  -  tan  9°  .  sin  J (87) 

In  a  given  latitude,  Eq.  (86)  will  make  known  the  shortest  twilight, 
and  Eq.  (87)  the  season  at  which  it  will  occur. 


*  Ann  Arbor  Astronomical  Notices,  N  .'I. 


76 


SPHERICAL    ASTRONOMY. 


§  308.  The  sign  of  the  second  member 
of  Eq.  (87)  shows  that  at  the  time  of 
shortest  twilight  the  spectator  and  the  sun 
will  be  on  opposite  sides  of  the  plane  of 
the  equinoctial. 

§  309.  The  depression  of  the  lowest 
point  Q'  of  the  equinoctial  below  the  ho- 
rizon HH',  is  90°  —  I  ;  and  of  the  low- 
est point  S  of  the  sun's  diurnal  path, 
when  his  declination  is  of  the  same  name 
as  the  spectator's  latitude,  90°—  (I  +  d)  ; 

and  when 

90°- 


=  18°, 

the  end  of  the  evening  will  be  the  beginning  of  morning  twilight,  and  the 
nocturnal  path  of  the  spectator  will  be  tangent  to  the  inner  boundary  of 
the  crepuscular  zone. 


THE  SUN. 

§  310.  The  Sun,  as  before  stated,  is  the  central  body  of  the  solar  sys- 
tem, and  from  this  circumstance  gives  to  the  latter  its  name.  It  occupies 
one  of  the  foci  of  all  the  elliptical  orbits  of  the  planets,  and,  of  course,  that 
of  the  earth. 

§  311.  Distance  and  Dimensions  of  the  Sun. — Its  horizontal  parallax 
denoted  by  P,  and  apparent  semi-diameter  denoted  by  s,  vary  inversely 
as  the  earth's  radius  vector.  For  the  mean  radius  it  is  found,  §  113-6, 

P  =  8".6,  ands  =  16'  01".5; 
which  in  Eqs.  (28)  and  (29)  give 

w  206264".8 


=  P-p  =  P- 

16'  01".5  _ 

"8^6"  ~P 


"7T-  =  23984  '  P 


961".5 
8".6 


=  111.5  p 


(88) 


.     .       (89) 


From  Eq.  (88)  it  appears  that  the  mean  distance  of  the  earth  from  the 
sun  is  23984  times  the  earth's  equatorial  radius ;  and  from  Eq.  (89)  thai 
the  sun's  diameter  is  111.5  times  that  of  the  earth.  The  volumes  of  these 
bodies  are  as  the  cubes  of  their  diameters,  and  hence  the  volume  of  the 
sun  is  1384472  times  that  of  the  earth. 


THE    SUN.  77 

g  312.  If  the  equatorial  radius  p  be  replaced  in  Eqs.    (88)  and    (89) 
by  its  value  in  miles,  §    98,  we  find 

rn  =  95,043,800  miles, 
2d=±       882,000     "     ; 

that  is  to  say,  the  mean  distance  of  the  earth  from  the  sun  is,  in  round 
numbers,  about  95  millions  of  miles,  and  the  diameter  of  the  sun  is  882 
thousand  miles.  The  mean  distance  of  the  earth  from  the  sun  is  assumed 
as  the  unit  of  linear  dimensions  in  all  celestial  measurements. 

§  313.  Mass  of  Sun.  —  In  Analytical  Mechanics,  §  201,  we  find  the 
equation 


(89)' 


in  which  T  denotes  the  periodic  time  of  a  body  revolving  about  a  centre 
of  attraction,  a  the  mean  distance  of  the  body  from  the  centre,  if  the  ratio 
of  the  circumference  to  the  diameter,  and  k  the  attraction  on  a  unit  of 
mass  at  the  unit's  distance. 

Let  k  become  fjo  in  the  case  of  the  sun's  action  on  the  earth  ;  then  will 
T  become  the  sidereal  year,  and  a  the  semi-transverse  axis  of  the  earth's 
orbit,  and 

-^    ........       (90) 

and  for  the  action  of  the  earth  upon  the  moon 


in  which  jx'  denotes  the^  attraction  on  the  unit  of  mass  at  the  unit's  distance 
exerted  by  the  earth. 

Now  the  attractions  exerted  by  two  bodies  on  the  same  mass  at  the 
same  distance,  are  directly  proportional  to  their  masses  respectively  ;  and 
denoting  the  mass  of  the  sun  by  M,  and  that  of  the  earth  by  M'  we  have 

£-t-r     £.,  (92) 

M1  ~~  <//  ~~  T*  '  a'3 

But  in  Eqs.  (62)  and  (88) 

T  =  365d.25,  and  a  =  23984  .  p  ; 

and  we  shall  presently  see  that  the  moon  revolves  about  the  earth  once  in 
27.5  days,  at  a  mean  distance  of  60  times  the  equatorial  radius  of  the 
earth.  Making,  therefore, 


78  SPHERICAL   ASTRONOMY. 

T  =  27.5,  and  a!  =  60  .  p, 
and  substituting  above,  we  have 

•p  =  354936.  .. 

That  is,  the  sun  contains  354,936  times  as  much  matter  as  the  earth  ;  and 
as  the  common  centre  of  inertia  divides  the  line  joining  their  respective 
centres  of  inertia  into  two  parts,  which  are  inversely  proportional  to  their 
masses,  the  common  centre  of  inertia  of  the  sun  and  earth,  about  which 
both  bodies  would  describe  their  respective  orbits  were  they  undisturbed 
by  the  other  bodies  of  the  system,  is  but  267  miles  from  the  sun's  centre, 
or  about  ^olfth  Par*  °f  ^  own  diameter. 

§  314.  Denote  by  D  the  density  of  the  sun,  and  by  Fits  volume;  also 
by  D'  and  F',  respectively,  the  density  and  volume  of  the  earth  ;  then 
Analytical  Mechanics,  §  18, 

M=D  .  F, 


and  by  division 


and  substituting  the  ratio  of  the  masses  and  of  the  volumes  given  above, 

we  find 

D  =  0.2543  .  D'; 

so  that  the  sun  is  but  a  trifle  more  than  one-quarter  as  dense  as  the  earth. 
The  latter  is  known,  from  the  recent  experiments  of  Mr.  Francis  Baily,  to 
be  5.67,  the  density  of  water  being  unity  ;  and  this  value  substituted  for 
D'  above,  makes  the  density  of  the  sun  not  quite  once  and  a  half  that  of 
water. 

§  315.  Surface  Gravitation  of  the  Sun.  —  By  the  laws  of  gravitation, 
the  attraction  of  one  body  upon  another  varies  as  the  quantity  of  matter  in 
the  attracting  body  directly,  and  the  square  of  the  distance  through  which 
die  attraction  is  exerted,  inversely.  The  distance  is  that  between  the  cen- 
tres of  gravity  of  the  bodies. 

Denote  by  W  and  W  the  weights  of  the  same  body  on  the  surfaces  of 
the  sun  and  earth,  respectively  ;  then  will 

W.   W  ::-  :  ^; 

e/a         p'  1 

whence  =>'  .....       (93) 


THE    SUN.  79 

and  substituting  the  values  just  found, 

W 

^  =  28,5. 

That  is,  a  body  weighing  one  pound  at  the*  equator  of  the  earth  \*culd 
weigh  28,5.  pounds  at  that  of  the  sun;  and  acquire,  therefore,  during  each 
second  of  its  fall  a  velocity  of  916,44  feet. 

§  316.  Sun's  Rotation  and  Axis.  —  Through  the  telescope          Fi&-  6S- 
the  sun's  surface  often  exhibits  dark  spots  which  slowly 
change  their  places  and  figure.     They  cross  the  solar  disk 
from  east  to  west,  and  thus  reveal  a  rotary  motion  of  the 
sun  itself  from  west  to  east  about  an  axis. 

§  317.  To  find  the  time  of  rotation  and  the  position  of 
the  axis,  it  will  be  necessary  first  to  find  the  heliocentric 
longitudes  and  latitudes  of  the  same  spot  at  different  times. 
To  do  this,  let  S  be  the  sun's  centre,  E  that  of  the  earth, 
P  the  spot,  and  N  its  projection  upon  the  plane  of  the 
ecliptic  Maks 

I  —  heliocentric  longitude  of  the  earth  ; 

x=  "  "  "       spot; 

y  =  P  S  JV  =  heliocentric  latitude  of  spot  ; 
<3  =  P  EN  =  geocentric  latitude  of  spot  ; 

e  =  SEN  =  difference  of  geocentric  longitude  of  the  sun  and  the  spot, 
J  =  sun's  apparent  semi-diameter. 

Then  SP  sin  y  =  P  N  =  EP  sin  /3  =  SE  sin  /3, 

because  the  difference  between  EP  and  S  E  is  insignificant  in  comparison 
with  either  ;  whence 

SE          _       sin  (3 


Again 

SP  .  cos  y  :  EP  .  cos  ft  :  :  SN  :  NE, 

:  :  sin  e  :  sin  (/•— 
whence 

sin  e  .  cos  (3    EP 


sm      —  x  = 


cos  y 

sin  e   .  cos  j8  ^ 
sin  4  .  cos  y  ' 


and  replacing  cos  y  by  its  value, 

sin  e  .  cos  8 

sm  (/  —  x)  =  -    ____  ; 

2      -9' 


80  SPHERICAL   ASTRONOMY. 

or  for  logarithmic  computation, 

sin  e  .  cos  (3 
sin  (I  —  x)  = 


. 
-V/sin  (J  +  £)  .  sin  (^  -  /3) 

§  318.  Position  of  the  Suns  equator,  and  the  time  of  the  Sun's  rota- 
tion. Let  E  be  the  pole  of  the  ecliptic,  P  that  of  the  sun's  equator  ;  A. 
A',  and  A"  the  heliocentric  places  of  the 
same  spot  observed  at  three  different  times  ; 
and  let  E  A,  E  A  \  E  A"  ,  PA,  PA',  PA" 
he  the  arcs  of  great  circles.  The  first  three 
are  known  from  Eq.  (94),  being  the  helio- 
centric colatitudes  of  the  spot  ;  as  also  the 
angles  A  E  A'  ,  AEA",  and  A'  E  A"  from 
Eq.  (95),  being  the  differences  of  the  he- 
liocentric longitudes  —  all  deduced  from  ge- 

osurface  observations  of  the  spot's  right  ascension  and  declination,  §  152. 
All  the  sides  and  angles  of  the  triangles  AE  A',  AEA",  and  A'EA" 
may  be  found,  two  sides  and  the  included  angle  in  each  being  given  ; 
hence  the  sides  A  A',  A'  A",  and  A"  A,  and  the  angles  A,  A',  and  A',  in 
the  triangle  A  A'  A",  are  known.  Now  P  being  the  pole  of  the  sun's 
equator,  parallel  to  which  the  spot  revolves, 

PA  =  PA'=  PA"-, 
Make 

2S  =  A  +  A'  +  A"  =  2P  AR  +  2PA'  A  -f  2PA'  A" 

=  2PAR  +  2Af: 
whence 

PAR  =  S  -  A', 

and  PAR  becomes  known. 

If  PR  be  perpendicular  to  AA"  , 


then  in  the  right-angled  triangle  APR,  the  angle  at  A  and  the  side  AH 
being  known,  the  side  PA  is  computed  j  and,  finally,  in  the  triangle 
APE,  the  sides  AP  and  AE,  and  the  angle  EA  P  =  EAA"  -PAA" 
being  known,  P  E  is  computed. 

§  319.  The  arc  EP  is  the  heliocentric  colatitude  of  the  pole  of  the 
sun's  equator,  and  the  angle  AE  P,  added  to  the  heliocentric  longitude  of 
the  spot  at  A,  gives  its  heliocentric  longitude.  The  position  of  the  sun's 
equator  becomes,  therefore,  known.  The  heliocentric  latitude  and  longi- 
tude of  its  north  pole  at  the  beginning  of  the  present  century  were,  respec- 
tively, 82°  30'  and  350°  21'. 


THE    SUN. 


81 


Fig.  65. 


From  the  triangle  APR  the  angle  AP R  becomes  known,  the  double 
of  which  is  AP  A".  Then,  denoting  by  T  the  time  of  one  rotation,  and 
by  t  the  interval  between  the  observations  on  the  spot  at  A  and  A",  we 

have 

A  PA"  :  t  ::  360°  :  T] 

whence  Tis  known  to  be  about  25.325  days,  making  the  angular  velocity 
of  the  sun  around  its  axis  about  one  twenty-fifth  that  of  the  earth. 

From  this  motion  it  is  concluded  that  the  sun  is  flattened  at  its  poles. 

§  320.   Physical  constitution  of  Sun. — 
'ihe  study  of  the  solar  spots  has  led  to  inter- 
esting conclusions  in  regard  to  the  physical 
constitution  of  the  sun  itself.     The  spots  are 
transient  in  character,  variable  in  size,  shape 
and  number,  and  confined  to  two  compara- 
tively narrow  zones  parallel  to,  and  at  no 
great  distance  from  the  sun's  equator.    They 
appear  perfectly  black,  and  surrounded  by  a 
border  less  dark,  called  a  penumbra.     The 
black  part  and  penumbra  are  distinctly  de- 
fined  in   outline,  and  do  not  fade  the  one  into  the   other.          F1»-  66- 
Sometimes  this  penumbra  presents  two  or  more  shades,  and 
in  this  case  also  there  is  no  gradation,  but  well-marked  out- 
line, indicating  a  total  absence  of  blending. 

As  the  spots  move  towards  the  edge  ri&-  67- 

of  the  sun,  the  penumbra  on  the  inner  :: ^  "         -- 

side  gradually  contracts,  and  with  the 
black  spot  disappears  before  reaching 
the  boundary  of  the  disk ;  the  penum- 
bra on  the  outer  side  expands,  and  is 
the  last  visible  remnant  of  the  spot  as  it  passes  behind  the  sun.  At  its 
reappearance  on  the  opposite  edge  of  the  sun,  the  spot  exhibits  similar 
phenomena — the  penumbra  first  appears,  then  the  black  portion  on  its  in- 
ner side,  the  contraction  of  the  penumbra  in  width,  and  its  extension* 
around  the  black  till  the  latter  is  entirely  surrounded. 

This  is  precisely  the  appearance  that  would  be  presented  by  a  deep  pit 
or  excavation  with  a  dark  or  non-luminous  bottom.  The  rotation  of  the 
sun  would  bring  the  slanting  surface  leading  from  the  inner  edge  of  its 
mouth  more  and  more  in  the  direction  of  the  spectator  till  it  would  be  lost 
in  the  foreshortening,  the  inner  edge  would  presently  mask  the  bottom, 
and  the  surface  of  the  opposite  side  would  be  turned  so  nearly  perpendicu- 

6 


82 


SPHERICAL    ASTRONOMY. 


larly  to  the  line  of  sight  as  to  appear  broadest  just  before  passing  behind, 
at  disappearance,  or  at  reappearance,  to  the  front  of  the  sun. 

§  321.  The  spots  gradually  expand  or  contract,  change  their  figure, 
vanish,  and  break  out  again  at  new  places  where  none  were  before.     When 


Fig.  68. 


disappearing,  the  central  black  part  contracts  to  a  point  and  vanishes  be 
fore  the  penumbra ;  and  a  single  spot  is  sometimes  seen  to  break  up  ink 
two  or  more  smaller  ones. 

§  322.  A  circle  of  which  the  diameter  is  one  second  is  the  smallest  vis- 
ible area.  A  single  second  at  the  earth  is  subtended  at  the  sun  by  a  dis 
tance  of  461  miles,  and  the  area  of  the  least  visible  circle  on  the  sun's 
surface  is,  therefore,  167,000  square  miles.  A  spot  whose  diameter  was 
45,000  miles  has  been  known  to  close  up  and  disappear  in  course  of  six 
weeks,  thus  causing  the  edges  to  approach  one  another  at  the  rate  of  1000 
miles  a  day.  Many,  spots  distinctly  visible  have  been  observed  to  vanish 
in  a  few  hours,  indicating  a  degree  of  mobility  inconsistent  with  the  idea 
of  solids  and  liquids. 

§  323.  Light  proceeding  very  obliquely  from  the  surfaces  of  incandes- 
cent solids  and  liquids  is  always  polarized,  whereas  that  from  gases  under 
the  same  circumstances  is  not.  The  light  from  the  edge  of  the  solar  disk 


THE   SUN.  83 

leaves  the  surface  of  the  sun  iu  a  direction  nearly  coincident  with  the 
surface  itself,  and  yet  when  examined  by  the  usual  tests  exhibits  no  signs 
of  polarization. 

§  324,  The  luminous  part  of  the  sun  is  not  uniformly  bright,  but  pre- 
sents a  mottled  appearance,  and  immediately  about  the  spots  are  often 
seen  well-defined  and  branching  streaks,  called  facules,  brighter  than 
other  parts  of  the  surface ;  among  these,  spots  often  make  their  appear- 
ance. They  are  best  seen  near  the  border  of  the  disk. 

§  325.  The  brightness  of  the  solar  disk  sensibly  diminishes  towards 
the  borders ;  and  this  fact  has  given  rise  to  the  supposition  that  the  sun 
is  surrounded  by  an  atmosphere  not  perfectly  transparent,  and  of  great 
extent  above  the  luminous  envelope.  The  loss  of  light  towards  the  bor- 
ders would  result  from  the  greater  absorption  of  the  luminiferotis  waves 
in  consequence  of  traversing  a  greater  thickness  of  the  atmosphere  in 
that  direction. 

§  326.  The  moon,  of  which  an  account  will  be  given  presently,  is 
known  to  be  a  non-luminous,  opaque,  spherical  mass,  and  so  near  the 
earth  as  to  give  to  it  an  apparent  diameter  about  equal  to  that  of  the 
sun.  This  little  body  often  interposes  itself  so  as  completely  to  conceal 
the  sun  from  view,  producing  what  is  called  a  solar  eclipse.  At  the  in- 
stant of  greatest  solar  obscuration — that  is,  when  the  rnoon  completely 
covers  the  sun — red  protuberances  resembling  flames  of  fire  are  seen  to 
issue  apparently  from  the  edge  of  the  moon,  but  in  fact  from  that  of  the 
sun,  revealing  the  existence  of  intense  commotion  and  physical  changes 
about  the  surface  of  the  latter  body. 

§  327.  From  all  which  it  is  inferred  that  the  sun  is  an  opaque  solid,  cov- 
ered by  a  gaseous  envelope  of  well-defined  boundary  and  intense  luminosity, 
the  whole  being  surrounded  by  a  non-luminous  atmosphere  of  vast  extent. 

No  explanation  free  from  objection  has,  thus  far,  been  given  for  the 
solar  spots.  Some  have  supposed  them  to  arise  from  scoria  or  flakes  of 
incombustible  matter  floating  upon  the  sun's  surface;  while  others,  with 
perhaps  greater  reason,  have  attributed  them  to  temporary  openings  in 
the  photosphere  that  envelops  the  sun,  exposing  to  view  detached  por- 
tions of  his  solid  crust,  which  appear  black  from  contrast. 

But  it  must  not  be  inferred  from  this  that  the  solid  portion  of  the  sun 
is  regarded  as  non-lnmiuous.  Were  he  stripped  of  his  gaseous  coating, 
he  would  no  doubt  shine  with  diminished  but  yet  intense  brilliancy.  A 
piece  of  quicklime,  in  a  state  of  most  active  combustion  under  the  action 
of  a  compound  blowpipe,  is,  when  projected  upon  the  bright  part  of  the 
sun,  as  dark  as  the  darkest  part  of  the  spots* 

During  the  interposition  of  the  lunar  sc:een  between   the  sun   and  ;« 


84  SPHERICAL   ASTRONOMY. 

spectator  on  the  earth,  the  surrounding  landscape  takes  on  the  obscure 
illumination  produced  by  a  closing  evening  twilight,  and  the  temperature 
is  always  sensibly  depressed,  thus  corroborating  the  suggestions  of  other 
phenomena,  that  the  sun  is  the  great  source  of  light  and  heat  to  the  earth. 
But  light  and  heat  are  the  results  of  molecular  agitation.  What, 
then,  is  the  cause  of  that  perpetual  molecular  vibration  essential  to  the 
self-luminosity  of  the  sun  ?  The  solar  system  is  believed  to  have  resulted 
from  the  subsidence  of  a  vast  nebula;  the  planets  and  satellites  are  de- 
tached fragments  left  behind  in  the  progress  of  the  general  mass  towards 
the  centre ;  the  sun  itself  is  the  central  accumulation.  This  nebula 
must  have  extended  originally  far  beyond  the  orbit  of  Neptune,  the  ex- 
tenor  planet  now  known.  The  distance  of  this  planet  from  the  sun  is 
more  than  thirty  times  that  of  the  earth.  The  condensation  has  taken 
place  under  the  action  of  weight  impressed  upon  the  elements  by  their 
reciprocal  attractions  for  one  another.  The  living  force  with  which  so 
much  matter  would  reach  the  terminus  of  a  fall  necessary  to  transfer  it 
to  its  present  abode,  could  not  fail  to  impress  upon  the  condensed  mass 
the  most  intense  molecular  agitation.  This  agitation,  or  molecular  liv- 
ing force,  can  only  be  lost  through  the  agency  of  the  surrounding  me- 
dium which  diffuses  it  through  space;  and  the  loss  in  a  given  time  is 
determined  by  the  density  of  the  medium,  being  less  as  the  density  is 
less.  The  medium  which  pervades  the  planetary  space  is  so  attenuated 
?<s  to  offer  no  sensible  resistance  to  the  denser  bodies  that  move  through 
it,  nor  could  we  be  conscious  of  its  existence  at  all  but  for  the  almost 
inconceivably  small  amount  of  living  force  which  it  brings  from  the  sun 
to  impress  upon  us  the  sensations  of  light  and  heat.  A  process  so  slow 
would  require  countless  ages  to  bring  the  solar  molecules  to  rest,  and 
convert  the  sun  into  a  non-luminous  mass. 

PLANETS. 

§  328.  Let  us  now  resume  the  Planets.  As  before  remarked,  these 
bodies  move  in  elliptical  curves,  of  which  one  of  the  foci  of  each  is  at  the 
centre  of  the  sun.  A  spectator  on  the  earth  views  these  bodies,  therefore, 
from  a  station  far  removed  from  their  centre  of  motion,  and  even  from  the 
planes  of  their  orbits.  Hence,  their  co-ordinates  of  place,  measured  by 
the  aid  of  instruments,  are  affected  with  both  geocentric  and  heliocentric 
parallaxes.  To  eliminate  these,  and  then  from  the  resulting  heliocentric 
co-ordinates  to  determine  the  elements  of  a  conic  section  whose  curve 
shall  pass  through  the  observed  places  and  have  a  focus  at  the  sun's  cen- 
tre, is  the  object  of  one  of  the  most  important  problems  in  Astronomy. 


Plate  H. 


TO  FBOJVT  PA.OE? 


PLANETS.  85 

Three  observed  right  ascensions  and  declinations,  together  with  the  inter- 
vals of  time  between  the  observations,  are  sufficient  for  its  solution. 

g  329.  The  planes  of  the  orbits  passing  through  the  sun,  the  orbit? 
themselves  will  pierce  the  plane  of  the  ecliptic  in  two  points,  called  jtodes. 
The  node  by  which  the  body  passes  from  the  south  to  the  north  of  the  eclip 
tic  is  called  the  ascending  node  /  the  other  is  calied  the  descending  node. 

§  330.  The  angle  which  the  plane  of  a  body's  orbit  makes  with  thai 
of  the  ecliptic  or  equinoctial,  is  calied  the  inclination. 

§  331.  The  semi-transverse  axis,  called  the  mean  distance,  and  eccen- 
tiicity,  determine  the  size  and  shape  of  the  conic  section. 

§  332.  The  inclination,  heliocentric  longitude,  or  light  ascension  of  the 
ascending  node,  and  of  the  perihelion,  fix  the  position  of  the  orbit  in  space. 

§  333.  The  time  of  the  body's  being  at  perihelion,  and  its  mean  angu- 
lar velocity,  called  its  mean  motion,  give  the  circumstances  of  the  body^s 
motion  in  the  orbit 

§  334.  The  orbit  of  a  heavenly  body  is  therefore  completely  deter- 
mined when  the  inclination,  mean  distance,  eccentricity,  longitude  of  the 
ascending  node,  longitude  of  the  perihelion,  epoch  of  the  perihelion  passage, 
and  mean  motion  are  known.  These  are  called  the  elements  of  an  orbit. 
They  are  seven  in  number. 

§  335.   To  find  a  plane  fs  elements.  —  The  polar  equation  of  the  orbit  is 


in  which  r  is  the  radius  vector  of  the  planet,  a  the  semi-transverse  axis, 
called  the  planet's  mean  distance,  e  the  eccentricity,  and  v  the  planet's  an 
gular  distance  from  perihelion,  called  the  true  anomaly  ;  the  pole  being 
at  the  sun. 

Making  v  =  90°,  r  becomes  the  semi-parameter,  which  denote  by  L, 
and  we  have,  Eq.    (96),  and  Analyt.  Mechanics,  §  200, 

L  =  a(l-e*)  =        -      .......     (97) 


in  which  c  denotes  the  area  described  by  a  radius  vector  in  a  unit  of  time  ; 
and  Eq.   (96)  may  be  written 

r  =  -  -  -  .....     (98) 

1  +  e  cos  v 

whence  e  cos  v  =  —  —  1      .     .     .     ..     .     .     .     (99) 

from  which,  denoting  the  planet's  velocity  in  the  direction  of  the  radius 
vector  by  Vn  we  find,  Appen  lix  VI., 


SPHERICAL    ASTRONOMY. 


es\uv  =  —  -.  Vr     ........  (100) 


which  divided  by  Eq.  (99)  gives 


§  336.  Denote  by  /?,  the  perihelion  distance,  then,  making  v  =  0,  in 

-q.    (96), 

p  =  a(l  -e) (102) 

§  337.  Denoting  by  T  the  periodic  time,  we  have,  An.  Mec.  §  201, 

.    ;        r=^;./.;;.t:.,.:.  .(1o8> 

V& 

and  denoting  the  mean  mati<m'\yy  », 


§  338.  Take  an  auxiliary  angle,  such  that 

cos  u  —  e 

eosv  = 

1  —  e  cos  1* 

•hen,  Appendix  VIL,          n  1  =  n  —  e  sin  u 

in  which  t  denotes  the  time  from   perihelion,  and  w,  as  above,  the  mean 

motion. 

§  339.  The  product  n  t  is  the  angular  distance  which  the  planet  would 
be  from  perihelion  had  it  moved  from  that  point  with  its  mean  motion  ny 
and  is  called  the  mean  anomaly. 

§  340.  The  auxiliary  angle  u  is  called  the  eccentric  anomaly,  and  dif- 
fers from  n  t  only  because  of  the  eccentricity  of  the  orbit ;  for  if  the  latter 
be  zero,  n  t  will  equal  u. 

§  341.  From  Eq.  (105)  we  readily  find 


§  342.  Making  in  Eq.  (103),  £=  fx,  a  =  1,  and  T=  365ll.256,  we  find 

log  it,  =  6.4711640 
log  VjL  —  8.2355820 

§  343.  From  the  centre  of  the  sun  draw  right  lines  respectively  to  the 
vernal  eq  linox,  intersection  of  the  solstitial  colure  with  the  equinoctial, and 
north  ce'estial  pole,  and  take  these  as  the  axes  ar,  y,  and  z.  The  planes 
of  the  equinoctial,  of  the  equinoctial  colure,  and  of  the  solstitial  coluro, 
^ill  be  the  co-ordinate  planes  x  y,  x  z,  and  z  y  respectively. 


PLANETS. 


87 


Denote  by  F^  Fy,  and  Vz  the  con:  ponent  velocities  of  the  planet  in  the 
direction  of  the  axes,  and  by  c',  c",  and  c'"  the  projections  of  c  on  the  co- 
ordinate planes  xy,zy,  and  z  x  respectively  ;  then,  Analytical 
§  189,  equations  (260),  will 


and 


ca  =  c'2  +  c"2  +  c' 


(108) 


(109) 


§  344.  Denote  the  inclination  of  the  orbit  to  the  plane  of  the  equinoc- 
tial by  »,  then  will 

(no) 

•     •' (HI) 


§  345.  Also, 
and,  Appendix  VIII., 


cos  ^  =  — 
c 


y-.V,  +  Z-.V, (112) 


Fig.  70. 


§  346.  Let  S  be  the  sun, 
P  the  place  of  the  planet,  R 
that  of  the  perihelion,  B  the 
vernal  equinox,  E  the  summer 
solstice,  A  the  north  celestial 
pole,  B  #the  ecliptic,  RP'N' 
the  intersection  of  the  plane  of 
the  planet's  orbit  with  the  ce- 
lestial sphere,  N1  the  heliocen- 
tric place  of  the  ascending 
node  N  on  the  equinoctial, 
A  P'P'"  and  A  R'R"  quad- 
rants of  great  circles  of  the  ce- 
lestial sphere.  Make 

X  =  B  P'"=  the  planet's 
heliocentric  right  as- 
cension. 

S  =.  A  P'  =  the  planet's  heliocentric  north  polar  distance. 

7j  =  N'P'"=  distance  in  heliocentric  right  ascension  from  the  node. 

s  =  B  Nf  =  heliocentric  right  ascension  of  ascending  node. 

<p  =  NP'  =  distance  of  the  planet  from  the  node. 
>/  =  N'R"  =  distance  of  perihelion  in  right  ascension  from  the  asc.  node. 

vt  =  B  R"  =  heliocentric  right  ascension  of  the  perihelion. 


88 
Then 


SPHERICAL    ASTRONOMY. 


tan  X  = 


(113) 


tan  P"SB  =  tan  P'QP"  -  cot  P' C A  =  -; 

x 


and  in  the  triangle  A  P'  C,  the 
side  AC  being  90°, 

cot  d  =  cosX-  -     (114) 

Again,  in  the  triangle  P'P"W, 
right-angled  at  P'", 

sin  v\  =  cot  8  -  cot  i  (115) 

e  =  X  — •»)         (116) 

tan  9  =  sec t* tan (X — s)    (117) 

In  the  triangle  ITR'R",  right- 
angled  at  72", 

tanX'=  cos  t.tan  (<p-f  0)  (118) 
in  which  v  is  the  true  anomaly 
P'SRf ;  and  hence 

tf  =  X'  +  s       (119) 

§  347.  It  thus  appears  that  as  soon  as  x,  y,  z,  FB,  Vy  and  Ve  are 
found,  all  the  elements  become  known  ;  and  the  preceding  formulas, 
arranged  in  the  order  of  sequence,  will  stand 


••  =  x^  +  yt  +  z1; 


(2) 


2c'"=  zVx—  xVt\ 
ca=  c'*+c"2+c'm. 

L=^, 


r       T      *      r      v     r      f* 
.     .    .     .    tuiv  -  —  .^—  -Ff. 


2c   Z- 


cos  v      r 


89 


PLANETS. 

/0x                                               r  (1  +  «  cos  v) 
(8) a  =  — ^— ^: 

(9) p  =  a  (1  -  e). 

(io) »  =  :4-' 

02 

/,,\  cos  v  -\-  e 

(1J)    .          ...     cos  u  = ;  Eq. 

l-J-6  cosv 

,  u  —  esinu 

(12) t  =  -  -;  Eq. 

(13) cos  t  =  — 

^  c 

(14) tanX=:y. 

•t 

(15) cot  5  =  cosX.-. 

(16) sin  r\  —  cot  5  .  cot  t. 

(17) e  =  X-n. 

(18) tan  <p  =  sec  t .  tan  (X  —  s). 

(19)  .....  tan  X'=  cose,  tan (<p+v). 

%  (20) tf  =  X'+  s. 

For  the  method  of  finding  x,  y,  z,  Vm  Vy,  Vt,  see  Appendix  IX. 

§  348.  The  sign  of  c'  in  Eq.  (110)  determines  the  inclination  to  be 
acute  or  oltus?,  and  also  the  direction  of  the  motion,  the  latter  being 
direct  when  c'  is  positive,  and  retrograde  when  negative.  The  planet 
will  be  receding  from  or  approaching  the  equinoctial  according  as  z  and 
Vz  have  the  same  or  opposite  signs,  and  it  will  be  north  or  south  of  the 
equinoctial  according  as  z  is  positive  or  negative.  The  signs  of  x  and 
y  will  show  in  which  quadrant  the  planet  is  projected  on  the  plane  of 
the  equinoctial.  See  Appendix  I. 

§  349.  The  position  of  the  orbit  is  given  in  reference  to  the  equinoc- 
tial ;  to  obtain  it  in  reference  to  the  ecliptic  is  a  mere  operation  of 
spherical  trigonometry  too  obvious  to  require  explanation. 


90  SPHERICAL   ASTRONOMY. 

§  350.  The  disturbing  action  of  the  planets  upon  one  another  causes  the 
nodes,  inclinations,  eccentricities,  and  perihelions  to  vary.  The  mean  rate 
of  change  in  each  case  is  found  by  dividing  the  whole  change,  as  ascer- 
tained at  epochs  widely  separated,  by  the  interval. 

§  351.  The  periodic  time  in  mean  solar  days  is  found  by  multiplying 
the  tabular  periodic  time,  which  is  expressed  in  that  of  the  earth  as  unity, 
by  365.24 ;  and  the  mean  distance  in  miles  will  be  given  by  the  product 
of  the  tabular  distance  into  95,000,000. 

§  352.  Dimensions  and  Geocentric  Distances. — Denote  by  X,  Y,  and  Z 
the  co-ordinates  of  the  earth ;  by  #,  y,  and  2,  those  of  a  planet,  referred  to 
the  centre  of  the  sun ;  and  by  D  the  distance  of  the  planet  from  the  earth. 
Then 


-zf      ....     (120) 

§  353.  The  horizontal  parallax  of  any  body  is  the  apparent  semi-diame- 
ter of  the  earth  as  seen  from  the  body.  Let  #  be  the  horizontal  parallax 
of  the  sun,  P'  that  of  the  planet,  and  r  the  radius  vector  of  the  earth ; 
then,  as  the  apparent  semi-diameter  of  the  earth  is  inversely  proportional 
to  the  distance  from  which  it  is  viewed,  will 


whence 

p'=*-i <121> 

and  this  in  Eq.  (29)  gives 

rf-P-^7 (122) 

in  which  s  is  the  planet's  apparent  semi-diameter  measured  with  the  mi- 
crometer, d  its  real  semi-diameter,  and  p  the  earth's  equatorial  radius ; 
whence  the  diameter,  surface,  and  volume  of  the  planet  become  known. 

§  354.  MERCURY  and  VENUS  are  called  inferior  planets,  being  lower 
or  nearer  to  the  sun  than  the  earth;  the  others  are  called  superior 
planets,  because  they  are  higher  or  more  distant  from  the  sun  than  the 
earth. 

§  355.  When  the  geocentric  longitude  of  a  body  is  the  same  as  that  of 
the  sun,  the  body  is  said  to  be  in  conjunction;  when  its  longitude  differs 
by  180°,  in  opposition.  The  superior  planets  may  be  in  opposition,  but 
the  inferior  planets  never. 

§  356.  A  body  in  conjunction  or  opposition  is  also  said  to  be  in  syzygy. 


PLANETS. 


Fig.  Ti. 


. 


§  357.  When  an  inferior  planet  is  in  perigean  syzygy,  it  is  said  to 
be  in  inferior  conjunction  ;  when  in  apogean  syzygy,  in  superior  con- 
junction. 

§  358.  Synodic  revolution.  —  The  interval 
of  time  between  two  consecutive  returns  of  a 
planet  to  apogean  or  perigean  syzygy  is 
called  its  synodic  revolution. 

Denote  by  m  the  heliocentric  mean  daily 
motion  of  the  earth  in  longitude  ;  by  TI,  that 
of  any  planet  ;  and  by  T,  the  length  of  its 
synodic  revolution  ;  then  will  m  ~  n  be  the 
relative  motion  in  longitude  of  the  earth  and 
planet,  and 


§359.  Geocentric  Motion  in  Longitude. — 
The  angle  at  the  earth,  subtended  by  a  body's 
linear  distance  from  the  sun,  is  called  the 
body's  elongation  ;  the  projection  of  a  body's 
centre  on  the  plane  of  the  ecliptic,  is  called 
the  reduced  place  ;  and  the  projection  of  its 
radius  vector,  is  called  the  curtate  distance. 

Thus,  let  S  be  the  sun,  P  a  planet,  E  the 
earth,  and  P  N  a  perpendicular  from  the 
planet  to  the  plane  of  the  ecliptic,  intersecting 
the  latter  in  N;  then  will  SEP  be  the 
elongation,  N  the  reduced  place,  and  S  N 
the  curtate  distance  of  the  planet. 

§  360.  Draw  S  V  and  E  V  to  the  vernal 
equinox  ;  they  will  be  sensibly  parallel.     Also 
drawJVA7/   and  E  Et  perpendicular  to  E  V   3 
and  S  F",  and  make 


a  =  S ~N  =  mean  curtate  distance  ; 
p  =  EN  =  earth's  distance  from  the  reduced  place ; 
/  =  VS  JV=  planet's  heliocentric  longitude ; 
n  =  hourly  change  in  the  same ; 

L  =  V  S  E  =  earth's  heliocentric  longitude ; 
X  =  V EN=  planet's  geocentric  longitude; 
m  —  hourly  change  in  the  same. 


92  SPHERICAL    ASTRONOMY. 

Then,  the  mean  distance  of  the  earth  from  the  sun  being  unity,  will. 
Appendix  X., 

m  =  Pz  [a2  +  a2  _  (a  +  a?)  .  cos  (L  —  1)]  .  n  .     .     .  (124) 

m  which 

cos  X 


p  — 


a  cos  I  —  cos  L 


and  which  will  make  known  the  rate  and  direction  of  the  body's  motion 
in  geocentric  longitude. 

§  361.  Direct  and  Retrograde  Motion  ;  Stations.  —  When  the  planet  is 
in  apogean  syzygy,  then  will  L  —  I  =  180°,  cos  (L  —  1)  =  —  1  ;  and, 
Eq.  (124), 

m  =  P2.a.(a  +  I)  (1  +a*).n  .....  (125) 

and  m  will  always  be  positive  ;  that  is,  the  geocentric  motion  of  the  planet 
will  be  direct. 

§  362.  When  the  planet  is  in  perigean  syzygy,  then  will  L  —  I  =  0  ; 
cos  (L  —  1)  =  1  ;  and,  Eq.  (124), 


a).n  ......  (126) 

and  m  will  always  be  negative,  whether  a  be  greater  or  less  than  unity  ; 
that  is,  the  geocentric  motion  of  the  planet  will  be  retrograde. 

§  363.  In  changing  from  direct  to  retrograde,  and  the  converse,  the 
body  must  appear  stationary.     This  will  make  m  =  0,  and,  Eq.  (124), 

coB(Z-J)  =  -  -  ^  =  _-    -I  --  ......  (127) 

1  +  «2       a2  +  a~  2  —  1 

a  quantity  which  is  always  less  than  unity,  whether  a  be  greater  or  less 
than  unity  ;  that  is,  all  the  planets  must  sometimes  appear  stationary. 
The  condition  expressed  by  Eq.  (127),  may  always  be  satisfied  for  two  val- 
ues of  L  —  I.  The  two  places  of  a  body,  in  which  it  appears  stationary, 
a  e  called  stations. 

§  364.  Let  the  value  of  L  —  I  for  one  of  the  stations  be  <p  ;  then,  Eq. 
(124), 

0  =  P2  [a2  +  a%  -  (a  +  a?)  cos  <p]  .  n  ; 
and  subtracting  from  Eq.  (124), 

m  =  P2  .  (a  -f  a^)  [cos  <p  —  cos  (L  —  /)]  .  n  .     .     .     (1  28) 

in  which,  as  long  as  9  is  less  than  90°,  and  L  —  I  greater  than  9  and  les? 
than  360°  —  (p,  m  will  be  positive  ard  the  motion  direct. 


PLANETS. 


93 


§  365.  Denote  by  n'  the  earth's  mean 
motion  in  longitude;  then  will  n  ~  n'  be 
the  mean  relative  heliocentric  motion  of 
the  earth  and  planet ;  and  denoting  by  tr 
and  td  the  durations  of  the  retrograde  and 
direct  motions,  we  have 


Fig.  za 


tr  = 


2<p 


~  n' 


(129) 


360°  —  2 


=  -— ^  •     •     (130) 


n 


and  the  duration  of  the  direct  motion  will  be  the  longer. 

It  thus  appeai-s  that  in  the  course  of  o"ne  synodic  revolution  the  planets 
appear  sometimes  to  be  stationary,  then  to  move  forward  or  in  the  order 
of  the  signs,  then  to  be  stationary  again,  and  finally  to  move  backwards. 

§  366.  Phases  of  the  Planets. — A  body  illumined  by  the  sun,  and 
shifting  its  place  in  reference  to  the  sun  and  earth,  presents  to  the  latter 
different  appearances  at  different  times.  These  appearances  are  called 
phases. 

§  367.  To  find  the  phase 
of  a  globular  body,  let  S  be 
the  place  of  the  sun's  centre, 
E  that  of  the  earth,  and  P 
that  of  the  body;  ADCB 
a  section  of  the  body  by  a 
plane  through  E,  P,  and  S; 
a  plane  through  P  and  per- 
pendicular to  P  E  will  cut 
from  the  body's  surface  the 
section  A  M  CN,  which  de- 
termines the  hemisphere 

turned  towards  the  earth ;  and  another  through  P,  and  perpendicular  to 
SP,  will  give  the  section  MDNB,  which  determines  the  illuminated 
hemisphere  turned  towards  the  sun.  The  illuminated  lower  surface 
MCNBM  will  be  visible  to  the  earth,  and  its  projection  on  the  plane 
AMCN  will  give  the  shape  and  magnitude  of  the  phase.  The  projection 
of  the  semicircle  MBNM  will  be  a  semi-ellipse  MB'  NM,  of  which  the 
transverse  axis  is  equal  to  the  diameter  of  the  body ;  its  conjugate  will 
vary  with  the  angle  which  the  projected  plane  makes  with  that  of  projec- 
tion. The  phase  will  therefore  have  for  its  boundary  a  semicircle  on  the 


SPHERICAL    ASTRONOMY. 


side  towards  the  sun  and  a  Fie-  T4  bis- 

serai-ellipse   on    the   other, 

these    being  united  at  the 

extremities   of    a    common 

diameter.    When  the  phase 

is  concave  on  the  elliptical 

side,  it  is  called  crescent; 

when  convex,#$6ows  ;  when 

straight,  dichotomous  ;  and 

when  the  ellipse  becomes  a 

.semicircle,  full.     Make 

d  =  distance  EP\ 

a  =  apparent  area  of  the  semicircle  M CNM at  distance  unity, 

«'=     "      "  "  "  distanced; 

p=     "      v      phase  MCATJS'M 
e=     "      "       semi-ellipse  MB' NM      "        " 
&  =  angle  BPB'  —  S  P  E\  the  exterior  angle  of  elongation. 


p  =  a  —  c 

€  =  Of  COS 


Then 
but 

whence 

p  =  a'  (1  —  cos  0). 

The  apparent  diameter  of  the  body  varies  inversely  as  the  first,  and  the 
apparent  area  of  the  disk  as  the  second  power  of  the  distance  ;  whence 


substituted  above  gives 


/-  *\ 

=  —  (1  —  cos  ^)  =  -^  .  ver  sin 


(131) 


§  368.  The  orbits  of  the  principal  planeta  have  but  slight  inclinations 
to  the  ecliptic.  At  inferior  conjunction  of  the  inferior  planets,  the  exterior 
angle  of  elongation  will  therefore  approach  to  0°,  and  the  distance  will  be 
the  least;  at  superior  conjunction  the  exterior  angle  will  be  180°,  and  the 
distance  the  greatest.  In  the  first  position  the  planet  will  be  invisible,  in 
the  second  full,  and  between  these  limits  the  phase  will  pass  through  cres- 
cent, dichotomous,  and  gibbous,  with  a  continually  decreasing  diameter. 
From  superior  to  inferior  conjunction  the  same  phases  occur,  but  in  the 
reverse  order. 

In  the  case  of  the  superior  planets,  the  exterior  angle  of  elongation  ap- 


PLANETS. 


95 


proaches  to  180°  both  at  conjunction  and  opposition,  and  it  never  can  be 
as  small  as  90°.     The  phases  of  these  bodies  must,  therefore,  always  be 
either  gibbous  or  full ;  largest  in  opposition,  and  smallest  in  conjunction. 
If  S  be  the  place  of  the  sun ;  E  that  of  the  earth ;    F,,  F2,  &c.,  the 


Fig.  75. 


places  of  an  inferior,  and  M\,  Mz,  &c.,  those  of  a  superior  planet,  then  will 
these  latter  bodies  exhibit  the  appearances  represented  in  the  figure. 

§  369.  Transits,  Occultations,  and  Transit  Limits. — A  body  which  in- 
terposes itself  between  the  earth  and  some  other  body,  so  as  to  conceal  any 
portion  of  the  latter  from  view,  is  said  to  make  a  transit ;  the  masked 
body  is  said  to  be  occulted,  and  the  phenomenon  is  called  a  transit  or  an 
occultation,  according  as  we  refer  to  the  masking  or  masked  body. 

§  370.  The  nodal  lines  of  all  the  planets  lying  in  the  plane  of  the  eclip- 
tic, are  crossed  twice  a  year  by  the  earth.  If  at  the  time  of  crossing  the 
nodal  line  of  an  inferior  planet,  the  latter  be  in  or  near  inferior  conjunction, 
there  will  be  a  transit,  and  the  planet  will  appear  as  a  dark  circle  on  the 
solar  disk. 

§  371.  To  find  the  greatest  elongation  consistent  with  a  Transit. — 
Conceive  a  conical  surface  tangent  to  the  sun  and  earth.  When  the  planet 


96 


SPHERICAL   ASTRONOMY. 


at  inferior  conjunction  passes  wholly  or  in 
part  within  this  surface,  there  will  be  a 
transit  visible  from  some  place  on  the 
earth. 

Let  S  be  the  sun,  E  the  earth,  and  P 
the  planet  just  touching  the  conical  sur- 
face, of  which  A  B  and  A'  B'  are  sections,  by  a  plane  through  the  centres 
of  the  three  bodies.     Make 

S  E  A  =  S   =  sun's  apparent  semi-diameter ; 

TEP  =  d  =  planet's  apparent  semi-diameter ; 

EA  B  =  ie  =  sun's  horizontal  parallax ; 

ETB  =  <*'  =  planet's  horizontal  parallax; 

SEP  =  s    =  planet's  elongation  at  the  beginning  of  the  transit; 

tiien  will 

s  =  &  +  d  +  AET, 
but 

A  ET  =  *'-«, 
whence 

8  =  6  +  d  +  */  —  # (132) 

that  is,  when  the  elongation  of  an  inferior  planet  is  less  than  the  sum  of 
the  apparent  semi-diameter  of  the  sun  and  planet,  augmented  by  the  differ- 
ence of  their  horizontal  parallaxes,  there  will  be  a  transit  or  an  occultation 
of  the  planet,  according  as  its  horizontal  parallax  is  greater  or  less  than 
that  of  the  sun. 

§  372.  Let  EE'  be  Fis- 7T- 

an  arc  of  the  ecliptic, 
0  0'  an  arc  of  the  plan- 
et's orbit,  and  N  the 
node.  Parallel  to  0  0', 
and  at  a  distance  from 
it  equal  to  5,  draw  on 
either  side  a  line  cutting  the  ecliptic  in  S. 

Now,  if  at  the  time  of  inferior  conjunction  the  difference  between  the 
geocentric  longitudes  of  the  sun  and  node  be  less  than  S  N,  there  must  be 
a  transit ;  if  greater,  there  can  be  none.  The  distance  S  N  is  called  a 
transit  limit. 

To  find  its  value,  make 

S  NP  =  i  =  inclination  of  the  planet's  orbit; 
=  I  =  transit  limit ; 


PLANETS.  97 

then,  in  the  right-angled  triangle  S  P  JV", 

sin  s 
smf  =  -  —  r       .......     (133) 

sm  * 

The  value  of  s  is  variable,  being  a  function  of  the  radii  vectors  of  the 
earth  and  planet  at  inferior  conjunction.  The  inclination  i  is  also  slightly 
variable.  The  greatest  value  of  i  and  least  value  of  s  make  I  a  minimum 
limit  ;  the  least  value  of  i  and  greatest  value  of  s  make  I  a  maximum 
limit. 

§  373.  The  earth  returns  sensibly  to  the  same  place  of  the  heavens  at 
intervals  of  a  sidereal  year.  Any  entire  number  of  sidereal  years  which 
will  contain  the  synodic  revolution  of  a  planet  a  whole  number  of  times, 
will  bring  the  earth  and  planet  to  the  same  places  they  simultaneously  oc- 
cupied before,  and  if  a  transit  occur  at  one  node,  it  will  occur  at  the  same 
node  again  at  the  expiration  of  this  interval,  provided  the  node  be  not 
carried  by  its  proper  motion  beyond  the  transit  limit. 

§  374.  The  bodies  having  returned  to  the  places  they  previously  occu- 
pied, will  each  have  performed  a  whole  number  of  entire  revolutions,  and 
making 

n  =  the  number  of  the  earth's  revolutions  ; 
n'  =  "  "      planet's       " 

P  =  the  length  of  the  earth's  sidereal  year  ; 
P'-  "          "      planet's      "          " 

we  shal!  have 

nP  =  n'P', 

whence 

=       ......  •  •  <I34> 


If  P  and  P/  be  whole  numbers,  and  the  second  member  be  reduced  to  its 
simplest  terms,  the  numerator  will  be  the  interval  in  sidereal  years  between, 
the  consecutive  transits  at  the  same  node,  and  this  interval  will  be  constant. 

But  if  P  and  P/  be  not  whole  numbers,  then  will  the  numerators  of  the- 
approximating  fractions  of  the  continued  fraction,  which  give  the  values  of 
the  second  member  within  the  transit  limits,  be  the  variable  intervals,  in 
sidereal  years,  between  the  transits  at  the  same  node. 

§  375.  Masses  and  Densities  of  the  Planets.  —  The  masses  of  such  of  the 
planets  as  have  satellites  may  easily  be  found  by  the  process  of  §  313,  as 
Boon  as  the  periodic  time  of  the  planet  and  that  of  its  satellite  are  deter- 
mined by  observation.  But  for  such  as  have  no  satellites,  recourse  is  had 
to  a  different  process,  which  can  be  here  indicated  only  in  outline.  A 


98  SPHERICAL  ASTRONOMY. 

planet  undisturbed  by  the  action  of  the  others,  would  describe  accurately 
its  elliptical  orbit  about  the  common  centre  of  inertia  due  to  its  own  mass 
and  that  of  the  sun ;  and  from  the  elliptical  elements  already  described, 
its  future  places  are,  as  we  shall  see,  predicted  with  the  greatest  precision. 
Tha  difference  between  these  places  and  those  actually  observed,  give  the 
effects  of  the  disturbing  action  of  the  other  planets.  To  compute  these 
effects,  what  are  called  perturbating  functions  are  constructed  upon  the 
principles  of  mechanics.  The  masses  of  the  perturbating  or  disturbing 
bodies  enter  these  functions ;  and  from  the  observed  amount  of  perturb- 
ations the  value  of  the  masses  are  computed.  An.  Mec.,  §  203, 

§  376.  The  masses  and  volumes  being  known,  the  densities  result  from 
the  process  of  §  314. 

§  377.  Rotary  motions. — All  the  planets  whose  surfaces  exhibit  through 
the  telescope  distinct  marks,  are  found  to  have  a  rotary  motion  in  the  same 
direction  as  those  of  the  sun  and  earth,  viz.,  from  west  to  east. 

§  378.  Planetary  Atmosphere. — The  existence  of  an  atmosphere  about 
a  planet  is  indicated  by  the  apparent  displacement  it  occasions  in  the  geo- 
centric place  of  a  star  by  refracting  its  light,  when,  by  the  motion  of  the 
earth  and  planet,  the  latter  comes  near  the  line  of  the  star  and  observer. 

The  atmosphere  about  a  planet  is  in  fact  a  vast  spherical  lens,  of  which 
the  central  part  is  deprived  of  its  transparency  by  the  opaque  materials  of 
the  planet,  but  of  which  the  outer  portion  is  free  from  obstruction  and  acts 
upon  the  light  which  passes  through  it  with  an  energy  due  to  its  refractive 
power  and  density. 

The  height  of  the  atmosphere  is  inferred  from  the  greater  or  less  angular 
distance  between  the  star  and  planet  when  the  displacement  begins ;  and 
the  density,  which  must  be  regulated  by  the  same  laws  that  govern  the 
equilibrium  of  heavy  elastic  fluids  upon  the  earth,  from  the  amount  of  dis- 
placement. 

§  379.  In  detailing  the  physical  peculiarities  of  the  planets,  their  mean 
distances  MTU!  times  of  sidereal  revolutions,  although  contained  in  the  sy- 
noptical table  of  elements,  will  be  repeated ;  and  in  all  cases  in  which  di- 
mensions or  measures  are  given,  they  must  be  understood  as  expressed  in 
the  corresponding  elements  of  the  earth  as  unity.  Thus,  if  it  be  the  mean 
distance,  density,  volume,  solar  heat  and  light,  sidereal  day,  <foc.,  those  of 
the  earth  are  the  respective  units. 


MERCURY.  99 

MERCURY. 

* 

§  3£0.  Proceeding  or^ wards  from  the  sun,  Mercury  is  the  first  known 
planet.  His  mean  distance  is  0.3870985  ;  sidereal  year,  0.2408;  true  di- 
ameter, 0.398;  volume,  O.OC3  ;  mass,  0.1 75;  density,  2.78 ,-approaching 
that  of  gold ;  intensity  of  its  attraction  for  a  unit  of  mass  on  its  surface, 
called  surface  gravitation,  1,15  ;  solar  heat  and  light,  6.68  ;  time  of  rota- 
tion upon  its  axis,  called  sidereal  day,  1 .20833. 

The  eccentricity  of  his  orbit  being  large,  his  greatest  elongation  varies 
from  16°  12'  to  28°  48'.  The  latter  being  his  greatest  apparent  distance 
from  the  sun,  he  is  generally  lost  to  us  in  the  light  of  that  body,  and  it  is 
difficult,  therefore,  to  observe  him.  His  arc  of  retrogradation  varies  from 
9°  22'  to  15°  44'. 

§  381.  When  to  the  west  of  the  sun  he  rises  before,  and  when  to  the 
east  he  sets  after  that  luminary.  In  the  former  position  he  is  called  a 
rooming,  and  in  the  latter  an  evening  star. 

§  382.  The  sun  appears  nearly  seven  times  as  large  to  the  inhabitants 
of  Mercury  as  to  us ;  and  on  the  supposition  that  the  intensity  of  solar 
light  and  heat  varies  .inversely  as  the  square  of  the  distance,  the  solar  il- 
lumination and  temperature  on  Mercury  would  be  6.68  times  that  on  the 
earth,  as  above.  Heat  and  cold  are,  however,  but  relative  terms,  depending 
upon  physical  conditions  as  well  as  distance,  and  the  Mercurian  surface 
may  be  as  cold  as  the  earth's ;  the  frosty  summits  of  the  Himalayas  are 
nearer  to  the  sun  than  the  scorching  plains  of  Hindostan. 

§  383.  The  changes  of  seasons  on  Mercury,  depending,  as  they  do, 
upoji  the  inclination  of  his  axis  to  that  of  his  orbit,  which  has  not  been 
well  determined,  are  not  accurately  known.  If,  as  there  are  reasons  to 
believe,  this  inclination  have  any  considerable  value,  the  mutations  of  Mer- 
cury's seasons  must  be  very  great;  his  tropical  year  being  only  about  one- 
fourth  that  of  the  earth,  his  seasons,  if  they  follow  the  same  proportion, 
can  only  be  of  some  two  or  three  weeks'  duration. 

§  384.  Mercury's  nodes  are,  and  will  for  ages  continue,  in  that  part  of 
tue  ecliptic  which  the  earth  passes  in  May  and  November,  and  his  transits 
over  the  sun  must  occur  in  those  months.  His  periodic  time  =  87d.97, 
and  that  of  the  earth  =  365d.256,  in  Eq.  (134),  give  the  approximating 

fractions, 

7        13        33 

29 ''     54  ;     137 '' 

So  that  the  intervals  between  the  transits  which  may  be  expected  at  the 
same  node  are  seven,  thirteen,  &c.,  years.     The  great  inclination  of  Mercu- 


100  SPHERICAL    ASTRONOMY. 

ry's  orbil  makes  his  transit  limits,  Eq.  (l  33),  small,  and  the  abo\  e  interra* 
will  not  therefore  always  be  those  which  separate  the  actual  recurrence  « 
the  transits.  The  last  transit  occurred  at  the  ascending  node  in  1848 
the  next  will  occur  in  1861. 


VENUS. 

§  385.  Venus  follows  Mercury  in  the  order  from  the  centre.  Her  mean 
distance  is  0.7233317;  sidereal  year,  0.61 52;  true  diameter,  0.975 ;  vol- 
ume, 0.927 ;  mass,  0.885;  density,  0.95;  surface  gravitation,  0.93;  solar 
heat  and  light,  1.91  ;  sidereal  day,  0.97315. 

§  386.  Venus  is  the  brightest  of  the  planets,  her  light  being  of  a  bril 
liant  white,  and  at  times  so  intense  as  to  cause  a  shadow.  The  elongations 
of  her  stations  vary  but  little  from  29°.  Her  phases  are  finely  exhibited 
through  the  telescope.  The  southern  horn  of  her  crescent  varies  its  shape, 
being  alternately  sharp  and  blunt,  and  the  changes  are  attributed  to  the 
periodical  interposition  of  high  mountains  by  an  axial  rotation  of  Venus 
so  as  to  intercept  the  solar  light  she  at  other  times  reflects  to  the  earth 
from  her  southern  surface.  From  these  changes  her  sidereal  day  has  been 
determined. 

§  387.  Her  axis  is  inclined  to  that  of  her  ecliptic  under  an  angle  of 
75°,  thus  placing  her  tropics  at  the  distance  of  15°  from  her  poles,  and 
her  polar  circles  at  the  same  distance  from  her  equator.  Her  seasons  suc- 
ceed each  other,  therefore,  very  rapidly,  there  being  two  summers  and  two 
winters  in  each  of  her  annual  revolutions.  Her  atmosphere  resembles  in 
extent  and  density  that  of  the  earth, 

§  388.  Her  synodical  revolution  is  583.92  days.  Venus  is,  therefore, 
about  292  days  continuously  to  the  east,  and  as  long  to  the  west  of  the  sun. 
In  the  former  position  she  sets  after  the  sun,  and  is  called  an  evening  star ; 
in  the  latter,  she  rises  before  the  sun,  and  is  called  a  morning  star.  Her 
greatest  elongation  is  about  45°,  and  she  is  brightest  when  on  her  way 
from  the  east  to  the  west  of  the  sun,  and  at  an  elongation  on  either  side  of 
about  40°. 

§  389.  The  line  of  Venus's  nodes  lies  in  that  part  of  the  ecliptic  through 
which  the  earth  passes  in  June  and  December,  and  her  transits  occur  in 
those  months.  The  periodic  time  of  Venus  =  224d.7,  and  that  of  the 
earth  =  365d.256,  which,  in  Eq.  (134),  give  the  approximating  fractions, 

8       235        713 
13'     382'     1159' 


VENUS, 


101 


Fig.  78. 


and  the  transits  at  the  same  node  may  be  expected  at  inteivals  of  eight, 
two  hundred  and  thirty-five,  <fec.,  years.  Two  transits,  separated  by  an 
interval  of  eight  years,  will  occur  at  one  node,  and  then  at  the  opposite 
node  after  an  interval  of  one  hundred  and  five,  or  one  hundred  and  twenty- 
two  years,  between  the  last  of  the  first  pair  and  first  of  the  second  pair, 

As  astronomical  phenomena  the  transits  of  Venus  are  of  the  highest  im 
portance.  They  afford  the  best  means  of  ascertaining  the  sun's  horizontal 
parallax,  and  therefore  the  earth's  distance  from  the  sun,  and  the  dimen- 
sions of  the  solar  system,  expressed  in  terms  of  some  known  terrestrial 
measure. 

§  390.  The  principle  on  which  the  sun's  horizontal  parallax  is  found 
from  a  transit  of  Venus  may  be  thus  illustrated, 

Conceive  two  observers  situ- 
ated at  the  opposite  extremities 
A  and  B  of  the  earth's  diameter, 
which  is  perpendicular  to  the 
plane  of  the  planet's  orbit.  To 
the  observer  A,  the  planet  would 
appear  to  transit  the  sun's  disk  along  the  chord  m  n,  and  to  the  observer 
B,  along  the  chord  p  q,  being  the  intersections  of  the  solar  disk  by  two 
planes  through  the  portion  D  C  of  Venus's  orbit,  described  during  the 
transit,  and  each  observer.  A  third  plane  through  the  observers  and  Ve- 
nus's centre  would  cut  from  the  other  two  the  lines  A  a  and  Bb,  and  from 
the  sun's  disk  the  perpendicular  distance  a  b  between  the  chords.  Now, 
because  the  angle  A  V B  is  equal  to  the  angle  a  Vb,  A  B  will  be  to  a  b  as 
Venus's  distance  from  the  earth  is1  to  her  distance  from  the  sun;  that  is 
§  385,  as  1—0.723  :  0.723,  or  as  1  to  2.61  nearly^,  and  the  radius  of 
die  earth,  half  of  A  B  is  to  a  6,  as  1  to  5.22  nearly.  The  apparent  mag 
nitudes  of  two  objects,  viewed  at  the  same  distance,  being  directly  pro- 
]K>rtional  to  the  true  magnitudes,  the  ra- 
dius of  the  earth  viewed  at  the  distance 
of  the  sun,  in  other  words,  the  sun's  hori- 
zontal parallax,  is  equal  to  the  angular 
distance  between  the  chords  divided  by 
5.22. 

§  391.  The  relative  geocentric  motion 
of  the  sun  and  planet  into  the  obserred 
durations  of  the  transit  at  the  two  stations 
will  give  the  chords  m  n  and  p  q.  The 
chords  being  known,  as  also  the  apparent 


102  SPHERICAL    ASTRONOMY. 

semi-diameters   Sq  and  S  n,  the  distances  SatmASb  become  known, 
and  therefore  their  difference  a  6. 

§  392.  The  general  result  of  all  the  observations  made  on  the  transit 
of  17G9  gives  8''.5776  for  the  sun's  horizontal  parallax.  The  next  two 
transits  of  Venus  will  occur  on  Dec.  8th,  1874,  and  Dec.  6th,  1882. 

MARS. 

§  393.  Mars  is  the  first  of  the  superior  planets.  His  mean  distance  is 
1.5237;  sidereal  year,  1.8807;  true  diameter,  0.517;  volume  0.1380; 
density,  0.95 ;  equatorial  gravitation,  0.493  ;  solar  heat  and  light,  0.43  ; 
sidereal  day,  1.02694  ;  oblateness,  about  19 ;  and  the  inclination  of  his 
axis  to  that  of  his  ecliptic  30°  18'  10".8. 

§  394.  He  has  a  dense  atmosphere  of  moderate  height.  His  surface 
(Plate  II.,  Fig.  2)  exhibits  through  the  telescope  outlines  of  what  are 
deemed  to  be  continents  and  seas,  the  former  being  distinguished  by  a 
ruddy  color,  which  is  characteristic  of  this  planet,  and  indicates  an  ochry 
tinge  in  the  soil,  contrasted  with  which  the  seas  appear  of  a  greenish  hue. 

These  markings  are  not  always  equally  distinct ;  and  the  variation  is 
attributed  to  the  formation  of  clouds  and  mists  in  the  planet's  atmosphere. 
Brilliant  white  spots  sometimes  appear  at  that  pole  which  is  just  emerging 
I'rom  the  long  night  of  its  polar  winter,  and  are  attributed  to  extensive 
snow-fields  that  push  their  borders  to  an  average  distance  of  some  six  de- 
grees from  either  pole. 

PLANETOIDS. 

§  395.  Next  to  Mars  come  the  class  of  small  planets,  which,  on  account 
of  their  comparatively  diminutive  size,  are  called  planetoids.  Little  is 
known  of  them  beyond  their  orbit  elements,  but  they  are  interesting  on 
account  of  their  history  and  the  speculations  connected  with  their  discov- 
ery, which  began  with  the  present  century. 

,§  396.  If  the  mean  distance  of  Mercury  be  taken  from  the  mean  dis- 
tances of  the  other  planets,  the  remainders  will  form  a  series  of  numbers 
doubling  upon  each  other  in  proceeding  outward  from  the  sun.  To  this 
law  there  was  a  remarkable  exception  in  the  distance  between  the  orbits 
of  Mercury  and  Jupiter  as  compared  with  that  between  Mercury  and  Mars, 
the  former  being  so  large  as  to  require  the  interpolation  ot  another  body 
between  Mars  and  Jupiter. 

§  397.  Although  th  *  law  is  strictly  empirical  and  wholly  inexplicable 


PI  ate  HI. 


TOFKONT  -PAGE 


•    PLANETOIDS.  103 

a  priori  upon  any  known  physical  hypothesis,  yet  the  coincidence  was  so 
remarkable  as  to  induce  the  prediction  that  by  proper  search  a  planet 
would  be  found  in  the 'interpolated  place. 

§  398.  This  body  was  only  to  be  recognized  by  its  proper  motion.  1\ 
detect  this,  an  examination  of  the  telescopic  stars  of  the  Zodiac  was  com- 
menced, their  places  were  carefully  mapped,  and  on  the  first  day  of  the 
present  century,  the  prediction  was  verified  by  the  addition  of  Ceres  to  the 
system.  Her  mean  distance  is  2.76692,  and  the  hiatus  was  filled. 

§  399.  But  the  discovery  of  Ceres  was  soon  followed  by  that  of  Pallas, 
at  the  mean  distance  of  2.7728 — nearly  the  same  as  that  of  Ceres — and 
the  law  was  again  broken. 

§  400.  The  points  in  which  the  paths  of  the  new  planets  are  intersected, 
3n  either  side  of  the  sun,  by  the  line  common  to  the  planes  of  both  orbits, 
are  not  very  far  apart,  and  it  was  suggested  that  Ceres  and  Pallas  were 
but  fragments  of  a  larger  planet  that  once  -revolved  at  an  average  distance, 
and  which  had  been  broken  to  pieces  by  some  disruptive  force.  But  where 
were  the  other  fragments  ? 

§  401.  A  number  of  bodies  projected  in  different  directions  from  a  com- 
mon point,  would  each  describe  about  the  sun  an  hyperbola,  a  parabola,  or 
an  ellipse,  depending  upon  the  relations  between  the  velocity  of  projection 
and  the  intensity  of  the  sun's  attraction  upon  the  unit  of  mass,  and  in  the 
case  of  elliptical  orbits,  the  bodies  would,  abating  the  effects  of  the  pertur- 
bating  action  of  the  other  planets,  return  at  fixed  intervals  to  the  place  of 
departure. 

§  402.  The  opposite  points  of  the  heavens,  in  which  the  orbits  of  Ceres 
and  Pallas  approached  most  nearly  each  other,  were  therefore  regarded  as 
the  common  haunts  of  the  suspected  fragments,  and  the  places  especially 
to  be  watched,  to  delect  their  existence.  A  constant  scrutiny  of  these 
points,  and  diligent  revision  of  the  maps  of  the  zodiac,  have  resulted  in  the 
discovery,  to  the  present  time,  of  91  of  these  little  bodies. 

§  403.  The  mean  distances  of  the  planetoids  vary  about  from  2.2  to  3.6, 
and  periodic  times  about  from  3.3  to  6.9.  Their  small  size  makes  it  diffi- 
cult to  determine  their  true  dimensions,  the  diametor  of  the  saYne  individ- 
ual, as  given  by  the  best  authorities,  varying  from  0.02  to  0.20.  They 
exhibit  considerable  variety  of  color ;  some  have  shown  signs  of  possessing 
atmospheres,  and  those  who  regard  them  as  debris  of  a  single  body,  find 
evidence  of  an  angular  or  fraginental  figure  in  sudden  changes  of  illumina- 
tion, which  have  been  observed,  and  which  are  attributed  to  the  shifting 
of  their  bounding  planes  by  a  diurnal  or  axial  rotation. 


104  SPHERICAL    ASTRONOMY. 


JUPITER. 

§  404.  Jupiter  is  the  largest,  and  except  Venus,  which  he  sometimes 
surpasses  in  this  respect,  the  brightest  of  the  planets.  His  mean  distance 
is  5.202  ;  sidereal  year,  11.86;  diameter,  11,2  ;  volume,  1280.9  ;  mass, 
331.57  ;  density,  0.24  —but  little  greater  than  that  of  water;  equatorial 
gravitation,  2.716;  solar  heat  and  light,  0.037;  sidereal  day,  0.41376; 
oblateness,  20 ;  inclination  of  axis  to  that  of  his  ecliptic,  3°  5'  30". 

§  405.  The  disk  of  Jupiter  is  always  crossed,  in  a  direction  parallel  to 
his  equator,  by  dark  bands  or  belts,  presenting  the  appearance  indicated  in 
Plate  III.,  fii>'.  3,  which  was  taken  by  Sir  John  H^rscliel.  These  belts  are 
not  always  the  same,  but  vary  in  breadth  and  situation,  though  never  in 
direction.  They  have  sometimes  been  seen  broken  up  and  distributed  over 
the  whole  face  of  the  planet.  From  their  parallelism  to  Jupiter's  equator, 
their  occasional  variation  and  the  .appearance  of  spots  upon  them,  it  is  in- 
ferred that  they  exisj  in  the  planet's  atmosphere,  and  are  composed  of 
extensive  tracts  of  clouds,  formed  by  his  trade-winds,  which,  from  the  great 
size  of  Jupiter,  and  the  rapidity  of  his  axial  rotation,  are  much  more  de- 
cided and  regular  than  those  of  the  earth. 

§  406.  The  great  oblateness  of  this  planet  is  due  to  the  shortness  of  his 
sidereal  day,  and  its  amount  agrees  with  that  assigned  by  theory  to  give 
him  a  figure  of  fluid  equilibrium. 

§  407.  From  the  small  inclination  of  his  axis  to  that  of  his  ecliptic,  there 
can  be  but  little  variation  in  the  length  of  his  days  and  nights,  each  of 
which  is  less  than  five  of  our  hours ;  and  changes  of  seasons  must  be 
almost,  if  not  quite  unknown  to  his  inhabitants. 

§  408.  Jupiter  is  attended  in  his  circuit  about  the  sun  by  four  satellites 
or  moons,  which  revolve  about  him  from  west  to  east,  and  present  a  min- 
iature system  analogous  to  that  of  which  Jupiter  himself  is  but  a  single  in- 
dividual, thus  affording  a  most  striking  illustration  of  the  effects  of  gravi- 
tation and  of  distance  in  grouping,  as  well  as  shaping  the  courses  of  the 
heavenly  bodies.  These  satellites  will  be  noticed  under  the  head  of  Sec- 
ondary Planets. 

SATURN. 

§  409.  Saturn  is  the  next  in  order  of  size  as  he  is  of  distance  to  Ju- 
piter. His  mean  distance  is  9.538850  ;  sidereal  year,  29.46  ;  true  diam- 
eter, 9.982  ;  volume,  995.00;  mass,  101.068  ;  density,  0.102— little  more 
than  half  that  of  water;  equatorial  gravitation,  1.014;  solar  heat  and 


Plate  IV. 


TO  FRONT  PAGE  1O4 


Plate  V. 


TO  FHONT  PAGE  105 . 


SATURN.  105 

light,  0.011 ;  sidereal  day,  0.43701 ;  oblateuess,  25;  inclination  of  axis 
to  that  of  orbit,  26°  49',  and  to  that  of  our  ecliptic,  28°  11'. 

§  410.  Saturn  is  the  most  curious  and  interesting  body  of  the  system, 
being  attended  by  eight  satellites  or  moons,  and  surrounded  (Plate  IV.,  Fig. 
4),  according  to  some  authorities  by  two,  and  others  by  four,  broad  flat  and 
extremely  thin  rings,  concentric  with  each  other  and  with  the  planet. 

§  411.  The  dimensions  of  the  rings  and  planet,  arid  the  intervals  as 
given  by  the  advocates  of  but  two  rings,  are, 

tf  miles. 

Exterior  diameter  of  exterior  ring  ....     40.095  =  176,418 
Interior         "  "         "      .     .     .     .     35.289  =  155,272 

Exterior  diameter  of  interior  ring  ....     34.475  =  151,690 
Interior         "  "         "       .     .     .     .     26.668  =  117,339 

Equatorial  diameter  of  planet 17.991=    79,160 

Interval  between  the  planet  and  interior  ring       4.339  =    19,090 

Interval  between  the  rings 0.408  =       1,791 

Thickness  of  ring  not  exceeding 230 

§  412.  The  evidence  of  recent  observations  with  very  powerful  instru 
inents  seems,  however,  in  favor  of  a  division  of  the  outer  ring,  as  just  given, 
at  a  distance  less  than  half  its  width  from  the  exterior  edge,  and  of  the 
existence  of  a  dusky  ring  still  nearer  the  body  of  the  planet,  and  composed 
of  materials  partially  transparent,  and  possessing  but  feeble  powers  of  re- 
flection, resembling  in  these  particulars  a  shee^  of  water.  And  there  seem 
good  reasons  for  believing  that  the  rings  are  not  precisely  in  the  same 
plane. 

The  disk  of  the  planet  is  crossed  by  parallel  belts,  similar  to  those  ot 
Jupiter ;  these  are  supposed  to  be  due  to  Saturn's  trade-winds.  From  the 
parallelism  of  the  belts  to  the  plane  of  the  rings,  it  is  inferred  that  the 
planet's  axis  of  rotation  is  perpendicular  to  that  plane,  and  this  is  con 
firmed  by  the  occasional  appearance  of  extensive  dusky  spots  on  his  sur- 
face, which,  when  carefully  watched,  give  the  time  of  his  rotation  about 
an  axis  having  that  direction. 

§  413.  By  watching  the  different  shades  of  illumination  on  different 
portions  of  the  rings,  the  latter  are  found  to  complete  a  revolution  in 
their  own  plane  once  in  10h  32m  15',  thus  making  their  sidereal  day 
0.43906,  which  exceeds  that  of  the  planet  itself  by  0.00205. 

§  414.  That  the  rings  are  opaque  and  non-luminous  is  shown  by  their 
throwing  a  shadow  on  the  body  of  the  planet  on  the  side  nearest  the  sun, 
and  bv  the  other  side  receiving  that  of  the  planet  as  shown  in  the  figure. 


106 


SPHERICAL   ASTRONOMY 
Fig.  80. 


§  415.  The  axes  of  the  planet  and  rings  preserve  their  directions  un- 
changed during  their  orbital  motion.  The  plane  of  the  rings,  which  is 
inclined  to  that  of  the  ecliptic  under  an  angle  of  31°  19',  intersects  the 
latter  plane  in  a  line  which  makes  with  the  line  of  the  equinoxes  an  angle 
equal  to  167°  31',  so  that  the  nodes  of  the  ring  lie  in  longitudes  167°  31' 
and  347°  31'. 

§  416.  The  orbital  motion  of  the  planet  causes  this  intersection  to  oscil- 
late, as  it  were,  parallel  to  itself,  in  the  plane  of  the  ecliptic,  through  a 
distance  on  either  side  of  the  sun  equal  to  the  radius  vector  of  Saturn's 
orbit ;  and  the  period  of  a  semi-oscillation  is  one-half  of  the  planet's  pe- 
riod, or  about  15  years.  Within  this  period  the  plane  of  the  ring  must  pass 
once  through  the  sun,  and  from  once  to  thrice  through  the  earth,  depend- 
ing upon  the  initial  position  or  place  of  the  latter  when  the  trace  of  the 
plane  on  the  ecliptic  touches  the  earth's  orbit  at  the  time  of  nearing 
the  sun. 

§  417.  Thus,  let  S  be  the  sun,  EE'E"E'"  the  earth's  orbit,  P  P'  an 
arc  of  Saturn's  orbit  projected  upon  the  plane  of  the 
ecliptic,  P  E  and  P'  E"  the  traces  of  the  plane  of 
the  rings  on  the  same,  and  tangent  to  the  earth's 
orbit,  and  suppose  the  motion  of  the  earth  and  of 
Saturn  to  take  place  in  the  direction  indicated  by  the 
arrow-heads.  Draw  SB  parallel  to  P  E  and  P'  E", 
and  make 

r  =  S  P        =  the  mean  distance  of  Saturn  ; 
r'=  SE       =  "         "         "         of  earth; 
a  =  P  S  P'  =  the  angle  at  the  sun  subtended  by 
PP'  : 

then,  since  the  angle  P  S  B  =  S  P  E,  we  have 


Fig.  81. 


«  =7=  9^4  =  0.1082, 


whence 


a  =  12°  2', 


SATURN. 


307 


which  divicbd  by  2'  0".6,  the  mean  motion  of  Saturn,  gives  350.46  days, 
wanting  only  5.8  days  of  a  complete  year ;  that  is  to  say,  the  earth  de- 
scribes nearly  one  entire  revolution  in  the  time  during  which  the  earth's 
orbit  is  traversed  by  the  plane  of  the  ring. 

§  418.  The  rings  are  invisible  when  their  plane  passes  between  the  sun 
and  earth,  their  enlightened  face  being  then  turned  from  the  latter  body  ; 
and  the  interval  of  non-appearance  will  be  that  between  any  two  epochs 
at  which  the  plane  passes  the  sun  and  earth,  and  of  which  the  effect  of 
one  is  to  throw  these  bodies  on  opposite  and  the  other  to  restore  them  to 
the  same  side  of  this  plane. 

§  419i  If  the  initial  place  of  the  earth  be  at  E",  nearly  three  days  in 
advance  of  B",  then  will  the  plane  itself  pass  the  sun  and  earth  at  the 
same  time,  the  earth  being  at  B' ,  and  these  bodies  could  not  be  on  oppo- 
site sides  of  the  plane  of  the  rings  during  its  present  visit  to  the  earth's 
orbit.  If  the  initial  position  of  the  earth  be  at  E',  nearly  three  days  in 
advance  of  E,  it  will  be  at  E"  when  the  plane  passes  the  sun  ;  the  rings 
will  then  disappear,  and  continue  invisible  till  the  earth  meets  and  passes 
their  advancing  plane,  which  it  will  do  somewhere  in  the  quadrant  E" B' '; 
they  will  then  reappear,  and  continue  visible  for  the  next  fifteen  years.  If 
the  earth's  initial  place  be  at  E'",  some  days  in  advance  of  B',  it  will 
meet  and  pass  the  plane  in  the  same  quadrant,  the  rings  will  disappear  and 
continue  invisible  till  their  plane  is  overtaken  and  passed  again  by  the 
earth  somewhere  in  the  quadrant  E  B" ',  when  the  plane  passes  the  sun 
the  earth  will  be  in  the  quadrant  B"E",  and  the  rings  will  again  disap- 
pear, and  again  become  visible  only  when  their  plane  is  recrossed  by  the 
earth  in  the  quadrant  E"B'.  Thus,  with  this  initial  place,  the  earth  will 
cross  the  plane  of  the  rings  three  times  in  one  year,  and  there  will  be  two 
disappearances. 

§  420.  When  the  plane  of  the  ring  passes  through  the  sun,  the  edge 
of  the  ring  alone  is  enlightened,  and  can  only  appear  as  a  straight  line  of 
light  projecting  from  opposite  sides  of  the  planet  in  the  plane  of  his  equa- 
tor, and  parallel  to  his  belts.  This  phase  of  the  ring  has  been  seen,  but  it 
requires  the  most  powerful  telescopes;  and  from  the  fact  of  its  non-ap- 
pearance in  a  telescope  which  would  measure  a  line  of  light  one-twentieth 
of  a  second  in  breadth,  of  which  the  subtense  at  Saturn's  distance  is  230 
miles,  it  is  inferred  that  the  thickness  of  the  ring  cannot  exceed  this  latter 
dimension. 

§  421.  When  the  dark  side  of  the  ring  is  turned  to  the  earth,  the 
planet  appears  as  a  bright  round  disk  with  its  belts,  and  crossed  equato- 
rial ly  by  a  narrow  and  perfectly  black  line.  This  can  only  happen  when 


108 


SPHERICAL    ASTRONOMY. 


the  planet  is  less  than   6°  1'  from  the  node  of  his  rings.     Generally  the 
northern  side  is  enlightened  when   the  heliocentric  longitude  of  Saturn  is 
between  172°  32'  and  341°  30',  and  the  southern  when  between  353°  32 
and  161°  30'.     The  greatest  opening  occurs  when  the  heliocentric  longi- 
tude of  the  planet  is  77°  31'  or  257°  31'. 

URANUS 

§  422.  Uranus  is  one  of  the  more  recently  discovered  planets,  being 
only  recognized  as  a  planet  for  the  first  time  in  1781,  though  it  had 
often  been  seen  before  and  mistaken  for  a  fixed  star. 

Of  this  planet  nothing  can  be  seen  but  a  small  round  uniformly  illumi- 
nated disk  without  rings,  belts,  or  discernible  spots.  His  mean  distance  is 
19.18239;  sidereal  year,  84.01;  true  diameter,  4.36;  volume,  82.91; 
mass,  14.25  ;  density,  0.17  ;  equatorial  gravitation,  0.75  ;  solar  heat  and 
light,  0.003.  He  is  attended  by  six  satellites,  which  will  be  noticed 
presently. 

NEPTUNE. 

§  423.  Neptune  is  the  last  known  planet  in  the  order  of  distance,  and 
third  in  size.  Its  discovery  dates  only  from  1846,  though  its  existence  had 
been  suspected  from  certain  irregularities  in  the  motion  of  Uranus,  which 
could  only  be  attributed  to  the  disturbing  action  of  some  body  exterior  to 
itself. 

The  departures  of  Uranus  from  places  assigned  by  the  combined  action 
of  the  known  bodies  of  the  system,  and  certain  assumed  conditions  in  re- 
gard to  position  and  shape  of  orbit,  direction  of  motion,  and  mean  distance, 
rendered  highly  probable  by  analogy,  were  the  data  from  which,  by  the 
methods  of  physical  astronomy,  was  wrought  out  in  the  closet  in  Paris, 
the  place  of  a  new  planet  whose  disturbing  action  would  account  for 
the  unexplained  waywardness  of  Uranus.  The  result  was  sent  to  an 
observer  in  Berlin,  and  in  the  evening  of  the  very  day  of  its  receipt  in 
the  latter  city,  Neptune  was  added  to  the  known  system  by  actual 
observation.  It  was  found  within  52'  of  the  place  assigned,  and  its 
discovery,  in  all  its  circumstances,  must  ever  be  regarded  as  one  of  the 
greatest  triumphs  of  modern  science. 

§  424.  Neptune's  mean  distance  is  30.0367  ;  periodic  time,  164.6181 ; 
real  diameter,  4.5  ;  volume,  91.125;  mass,  18.219  ;  density,  0.208  ;  equa- 
torial gravitation,  0.9035  ;  solar  heat  and  light,  0.0011. 

The  apparent  size  of  the  sun  as  seen  from  the  earth,  bears  to  that  as  seen 


SECONDARY    BODIES.  109 

from  Neptune,  about  the  relation  of  an  ordinary  orange  to  a  common  duck- 
shot. 

§  425.  Neptune  has  at  least  one  satellite,  and  certain  appearances  have 
indicated  a  second,  and  also  a  ring,  but  of  these  there  are  yet  doubts. 

General  Remark. 

§  426.  In  the  foregoing  enumeration  of  the  physical  peculiarities  of  the 
planets,  one  is  impressed  by  the  great  differences  in  their  respective  sup- 
plies of  heat  and  light  from  the  sun ;  in  the  relations  which  the  inertia  of 
matter  bears  to  its  weight  at  their  surfaces ;  and  in  the  nature  of  the  ma- 
terials of  which  they  are  composed,  as  inferred  from  variety  of  mean 
density.  The  intensity  of  solar  radiation  is  nearly  seven  times  greater  on 
Mercury  than  on  the  earth,  and  on  Neptune  900  times  less,  giving  a  range 
of  which  the  extremes  have  the  ratio  of  6300  to  1.  The  efficacy  of  weight 
in  counteracting  muscular  effort  and  repressing  animal  activity  on  the 
earth,  is  less  than  half  that  on  Jupiter,  more  than  twice  that  on  Mars,  and 
probably  more  than  twenty  times  that  on  the  planetoids,  making  a  range 
of  which  the  limits  are  as  40  to  1.  Lastly,  the  density  of  Saturn  does  not 
exceed  that  of  common  cork.  Now,  under  the  various  combinations  of 
elements  so  important  as  these,  what  an  immense  diversity  must  exist  in 
the  conditions  of  animal  life,  if  the  planets,  like  our  earth,  which  teems 
with  living  beings  in  every  corner,  be  inhabited  !  A  globe  whose  surface 
is  seven  times  hotter  than  ours  or  900  times  colder,  on  which  a  man  might 
by  a  single  muscular  effort  spring  fifty  feet  high,  or  with  difficulty  lift  his 
foot  from  the  ground ;  where  his  veins  would  burst  from  deficiency  or  col- 
lapse from  excess  of  atmospheric  pressure,  affords  to  our  ideas  an  inhospi- 
table abode  for  animated  beings.  But  we  should  remember  that  heat  and 
cold,  light  and  darkness,  strength  and  weakness,  weight  and  levity,  are  but 
relative  terms ;  and  to  the  very  conditions  which  convey  to  our  minds  only 
images  of  gloom  and  horror,  may  be  adjusted  an  animal  and  intellectual 
existence  which  make  them  the  most  perfect  displays  of  wisdom  and  be- 
neficence. 

SECONDARY  BODIES. 

§  427.  The  secondary  bodies  are  those  which  revolve  about  the  planets, 
and  accompany  them  around  the  sun.  Of  these,  twenty  are  known  at  the 
present  time.  One  belongs  to  the  earth,  four  to  Jupiter,  eight  to  Saturn. 
six  to  Uranus,  and  one  to  Neptune.  They  are  commonly  called  satellites, 
and  sometimes  moons,  but  this  latter  appellation  is  more  particularly  ap- 
plied to  the  earth's  secondary. 


110 


SPHERICAL    ASTRONOMY 


THE  MOON. 

• 

§  428.  The  moon  revolves  in  an  elliptical  orbit,  of  which  one  of  the 
foci  is  at  the  earth's  centre.  Its  motion  is  from  west  to  east,  and  its  an- 
gular velocity  about  the  earth  is  much  greater  than  that  of  the  earth 
around  the  sun.  The  moon  appears,  therefore,  to  move  among  the  fixed 
stars  in  the  same  direction  as  the  sun,  but  more  rapidly  ;  and  from  the 
axial  motion  of  the  earth  she  has,  like  other  heavenly  bodies,  an  apparent 
diurnal  motion,  by  which  she  rises  in  the.  east,  passes  the  meridian,  and 
sets  in  the  west. 

§  429.  The  oblateness  of  the  earth  would  be  quite  appreciable  to  an  ob 
server  at  the  distance  of  the  moon.  Her  equatorial  horizontal  parallax  is 
therefore  found  from  Eq.  (24)  ;  her  distance  from  Eq.  (28)  ;  her  true  diam- 
eter from  Eq.  (29);  arid  her  mass  from  her  effects  in  producing  precession 
and  nutation. 

Lunar  Orbit. 

§  430.  The  elements  of  the  moon's  orbit  may  be  found  from  four  ob- 
served right  ascensions  and  declinations,  corrected  for  refraction,  parallax, 
and  semi-diameter. 

Let  D  C  be  an  arc  in  which  F1?  82- 

the  plane  of  the  orbit  cuts  the 
celestial  sphere  ;  V  B  an  arc  of 
the  ecliptic,  and  V  A  of  the 
equinoctial  ;  V  the  vernal  equi- 
nox, N  the  ascending  node,  P 
the  perigee,  and  J/,,  M.2,  M*. 
MI  the  geocentric  places  of  the 
moon. 

First  convert  the  geocentric 
right  ascensions  and  declina- 
tions into  geocentric  longitudes 
-md  latitudes,  and  make 

v  =  V  N     :==  longitude  of  node  ; 
i  =.  C  'N  B  ±=  inclination  of  orbit; 
li  =  V  0,     =  longitude  o 


X,  =  J/i  0,    a  latitude  of  Jf,  ; 


THE    MOON.  in 

then  in  the  right-angled  triangles  Jt/j  JV  0,  and  M2  N  08,  we  have 

sin  (I,  -  v)  =  cot  f  .  tan  X,  i 
sin  (4  —  v)  =  cot  *  .  tan  X2  ) 
And  by  division 

sin  (1L  —  v)       tan  X, 

sin  (12  —  v)       tan  X2* 

Adding  unity  to  both  members,  reducing  to  common  denominator,  then 
subtracting  each  member  from  unity,  reducing  as  before,  and  finally  divi- 
ding one  result  by  the  other,  we  find 

sin  (/2  —  v)  -f-  sin  (/,  —  v)       tan  X2  -f-  tan  X, 
sin  (72  —  v)  —  sin  (/,  —  v)  ~~  tan  X2  —  tan  X,  ' 

replacing  the  members  by  their  equals,  we  have 


Also,  from  first  of  Eqs.  (135),  we  have 

cot»  =  sin(V-^   .  (137) 

tan  X, 

whence  v  and  i  are  known. 

The   longitude  of  the  ascending  node,  increased  by  the  angular  dis- 
tance of  a  body  from  the  same  node,  is  called  the  Orbit  Longitude, 

Make 

v,  =  YEN  +  NEM,  =  orbit  longitude  of  Ml  ; 

p=VEN+NEP  —     "  "  perigee; 

<p  =  PJEMi  =  vl  —  p  =  true  anomaly  of  M\  ; 
e  =  eccentricity  of  orbit  ; 

m  =  mean  motion  of  moon  in  orbit  ; 

f,  =  time  since  epoch  for  Ml  ; 

L  =  mean  orbit  longitude  at  epoch. 

Then  resuming  Eq.  (48),  we  have 

L  +  mti  =  v,  —  1e  sin  (v,  —  p)      .     .     .     .     (138) 
in  which 

,1  =  v  +  tan-.^^)    ----     (,39) 

Four  values  for  the  geocentric  longitudes  denoted  by  /„  4>  I*  ^>  m  Eq. 
(139),  give  four  values  for  v,  viz.,  v,,  v,,  i>,,  and  v4  ;  and  these,  and  the  times 


SPHERICAL    ASTRONOMY. 

of  observation  h,  t^  t$,  and  £4,  in  Eq.  (138),  give  four  equations  involving 
the  four  unknown  quantities  Z,  m,  e,  and  p  ;  whence  these  become  known 
precisely  as  in  §  197,  employing  for  the  purpose  Eqs.  (50),  (51),  (53), 
and  (54). 

§  431.  Denoting  the  ecliptic  longitude  VO  of  the  perigee  by  p{,  we 
have,  in  the  triangle  NP  0,  right-angled  at  0, 

tan  N  0  =  tan  (p  —  v)  .  cos  i, 
and 

Pi  =  v  +  tan"1  [tan  (p  —  v)  .  cos  i]    .     .     .     (140) 

§  432.  In  the  same  way,  denoting  the  mean  ecliptic  longitude  of  the 
moon  at  the  epoch  by  L\, 

LI  =  v  +  tan'1  [tan  (L  —  v)  .  cos  i]     .     .     .     .     (141) 

§  433.  The  passage  of  the  moon  through  one  entire  circuit  of  360° 
around  the  earth,  is  called  a  sidereal  revolution.  The  interval  of  time  re- 
quired to  perform  a  sidereal  revolution  is  called  a  sidereal  period.  Denote 
the  sidereal  period  by  s,  then  will 

360° 
*  =  -  .........     (142) 

m 

The  equation  of  the  orbit,  the  centre  of  the  earth  being  the  pole,  is 


I  +  e  cos  (v  —  p)' 

and  the  value  of  r  being  found  by  means  of  Eq.  (28),  that  of  the  mean 
distance  a  will  result,  and  every  thing  in  regard  to  the  moon's  path  be- 
comes known. 

§  434.  At  the  epoch  January  1st,  1801,  the  elements  of  the  lunar  orbit 

were 

Mean  a  =  59.96435000  of  the  earth's  equatorial  radius; 

"      s  =  27.321661418  mean  solar  days; 
"      e=    0.054844200  ; 
_        "      v=    13°  53'  17".7; 

"  ^,  =  266°  10'  07".5; 

"  .»  =      5°08'47".9; 

"  Ll  =  118°  17'  08".3. 

g  435.  The  moon's  true  diameter,  Eq.  (29),  is  0.27280,  or  about  2153 
miles;  volume,  0.0204;  mass,  0.011399;  density,  0.5657;  and  surface 
gravitation,  0.1666. 

§  436.  Comparing  the  lunar  elements  which  depend  upon  the  orbit  aa 


THE    MOON 


113 


determined  at  different  times,  they  are  all  found  to  va,y.  The  nodes  have 
a  retrograde  and  the  perigee  a  cfTrect  motion,  the  former  performing  a  com- 
plete revolution  in  18.6,  and  the  latter  in  8.854  years.  The  inclination 
fluctuates  between  4°  57'  22"  and  5°  20'  06" ;  the  mean  distance  has  a 
secular  variation,  and  it  is  at  the  present  time  diminishing ;  the  same  is 
true  of  the  sidereal  revolution,  and  the  mean  motion  of  the  moon  is  increas- 
ing. All  these  changes  are  due  to  the  disturbing  action  of  the  other  bod- 
ies of  the  system,  but  principally  of  the  sun.  The  action  of  the  protuberant 
ring  of  matter  about  the  equator  of  the  earth  also  has  its  effect. 


Disturbing  Forces. 


Fi&- 


§  437.  To  illustrate  the  way  in  which 
these  changes  of  the  lunar  orbit  are 
brought  about,  let  E  be  the  earth,  S  the 
sun,  M  the  moon,  moving  in  her  orbit  in 
the  &\vwt\oi\MDN'N',  N  and  N'  be- 
ing the  nodes,  and  EV  the  direction  of 
the  vernal  equinox.  Then,  resuming 
Aquations  (80)  and  (81),  making  p  = 
EM,  the  radius  vector  of  the  moon,  and 
employing  in  all  other  respects  the  nota- 
tion of  §  286,  v  becomes  the  change 
which  the  sun's  attraction  causes  in  the 
weight  of  a  unit  of  the  moon's  mass  due 
to  the  earth's  attraction,  and  r  the 
change  which  the  sun's  attraction  causes 
in  the  force  normal  to  the  radius  vector 
and  in  the  plane  passing  through  the  sun, 
earth,  and  moon.  This  latter  force  being 
in  general  oblique  to  the  plane  of  the  lu- 
nar orbit,  urges  the  moon  out  of  that 
plane,  and  causes  her  to  describe  a  curve 
of  double  curvature,  while  the  former  has 
no  such  action. 

Resolve  T  into  two  components,  one  perpendicular  to  the  radins  vector 
and  in  the  plane  of  the  orbit,  the  other  normal  to  this  latter  plane.  For 
this  purpose  conceive  a  sphere  of  which  the  centre  is  at  that  of  the  earth, 
and  radius,  the  radius  vector  p  =  EM,  of  the  moon.  Its  surface  will  be 
cut  by  the  plane  of  the  ecliptic  in  ANtBNti,  by  that  of  th*  lunar  orlit 

8 


SPHERICAL    ASTRONOMY 


in  Nt  MNtl,  and  by  that  of  the  sun, 
earth,  and  moon  in  AM B.     Make 

O  =  V  E  S  =  sun's  longitude  ; 
ft  =  YEN,—  long,  of  moon's  node; 
i  =  MN4  A  =  inclination  of  lunar  or- 
bit; 

X  =±  NtMA  =  inclination  of  orbit  to 
plane  of  sun,  earth, 
and  moon ; 

*  =  <r  .  cos  X  =  component    of    r    in 
plane  of  lunar  orbit ; 
o  =  r  .  sin  X  =  component  normal  to 
plane  of  orbit. 

In  the  triangle  Nt  MA,  the  arc  Nt  A 
=  (0  —  ft)  and  A  M  =  9,  and  we  have 


Fig.  88  bia. 


sin   X  = 


sin  i  .sin  (0  —  ft) 

sin  9 

sin2  ^.  sin2  (O  —  ft)  ^ 


SOT 


and  bringing  forward  Eq.  (80),  and  replacing  r  by  its  value  in  Eq.  (81), 
there  will  result 

v  =  —  ^— £  .  (2  cos2  9  —  sin2  9) (143) 


(144) 


.  cos  9  .sinu  sin  (0  —  ft)    ....     (145) 


v  is  called  the  radial,  if  the  transversal,  and  o  the  orthogonal  disturbing 
force.  These  forces  acting  at  right  angles  are  independent  of  each  other, 
and  affect  the  movements  of  the  moon  in  modes  perfectly  distinct,  which 
will  become  manifest  by  discussing  the  above  equations. 

§  438.  The  radial  force  being  directed  towards  or  from  the  earth's  centre, 
changes  the  simple  law  of  gravitation,  and  alters  the  elliptic  form  of  the 
orbit,  sometimes  diminishing  and  sometimes  increasing  its  eccentricity, 
and  shifting  the  place  of  its  greatest  curvature,  that  is,  the  position  of  the 
apsides. 

§  439.    The  transversal  force  is  exerted  to  accelerate  or  retard   the 


THE    MOON.  }J5 

moon's  motion  in  her  orbit,  and  to  give  rise  to  fluctuations  to  and  fro 
about  that  due  to  the  action  of  the  earth  alone,  and  thus  to  alter  the  ellip- 
tic path. 

§  440.  The  orthogonal  force  deflects  the  moon  from  the  plane  in 
which  she  would  move  under  the  undisturbed  action  of  the  earth,  and 
causes  her  to  describe  a  curve  of  double  curvature.  A  plane  through  two 
consecutive  elements  of  such  a  path  must  in  general  be  oblique  to  that 
through  two  other  consecutive  elements,  and  these  two  planes  must  in 
general  intersect  the  plane  of  the  ecliptic  in  different  lines ;  that  is,  the 
orthogonal  force  is  effective  in  producing  a  motion  of  the  nodes.  By  dis- 
cussing the  value  of  this  force,  it  will  be  found  that  while  it  causes  the 
nodes  to  move  in  different  directions  at  different  times,  on  the  whole,  it 
Causes  them  to  retrograde. 

§  441.  Nothing  has  thus  far  been  said  of  the  variations  in  these  disturb* 
iug  forces  arising,  all  other  things  being  equal,  from  the  change  in  the 
value  of  d)  or  the  earth's  distance  from  the  sun.  This  gives  rise  to  a  still 
further  complication  by  introducing  an  annual  variation  in  the  values  of 
v,  <7r,  and  o. 

§  442.  The  other  bodies  of  the  system  produce  effects  similar  to  thos-s 
of  the  sun,  but  much  less  in  degree.  The  protuberant  ring  of  matte; 
*hi«~V  projects  beyond  the  sphere  described  upon  the  earth's  polar  axis 
aas,  as  already  remarked,  its  effect  also ;  so  that  the  longitude  of  the 
moon,  as-  Determined  from  the  elements  of  a  true  elliptic  motion,  must  re- 
ceive from  30  to  40  corrections  to  obtain  that  of  her  true  place. 

§  443.  These  coirections  are  called  equations ;  their  forms  are  deter- 
mined by  investigations  in  physical  astronomy,  and  their  coefficients  are 
computed  from  the  observed  departures  of  the  actual  from  the  elliptic 
places. 

Librutions. 

§  444.  The  moon  revolves  uniformly  about  an  axis  inclined  to  that  of 
her  orbit,  under  an  angle  of  6°  38'  58",  which  is  slightly  variable ;  and 
the  time  of  one  revolution  is  equal  to  her  sidereal  period. 

§  445.  Were  her  orbital  motion  uniform,  and  her  axis  perpendicular  to 
her  orbit,  this  equality  would  cause  the  moon  always  to  present  the  same 
face  to  the  earth.  As  it  is,  however,  the  visible  portion  of  her  surface  is 
slightly  variable,  and  in  the  course  of  a  sidereal  period  we  see  a  little  more 
than  a  hemisphere.  The  changes  of  orbital  motion  cause  small  portions  of 
her  surface  near  her  eastern  and  western  borders  to  enter  arid  depart  from 
the  field  of  view  in  the  course  of  each  revolution ;  and  the  inclination  of 


116 


SPHERICAL   ASTRONOMY 


her  axis  to  that  of  her  orbit  exposes  to  us  her  north  or  south  pole  alter 
nately  within  the  same  period.  These  circumstances  give  rise  to  appa- 
rent oscillatory  motions  in  the  moon  itself,  whiph  are  called  librations  ; 
those  due  to  irregnlarity  of  orbital  motion  are  called  longitudinal,  and 
those  which  arise  from  inclination  of  axis,  latitudinal  librations. 

§  446.  In  addition,  slight  variations  take  place  in  the  visible  portions 
of  the  moon's  surface  from  changes  in  the  observer's  point  of  view,  by  the 
earth's  rotation.  These  are  called  parallactic  lib-rations. 

Lunar  Periods, 

§  447.  The  moon's  equinoctial  is  inclined  to  the  ecliptic,  and  its  ascencfc- 
ing  node  always  exactly  coincides  with  the  descending  node  of  her  orbit; 
so  that  the  moon's  axis  describes  a  conical  surface  about  the  axis  of  the 
ecliptic  once  in  1  8.6  years, 

§  448.  The  passage  of  the  moon  from  conjunction  with  the  sun  to> 
conjunction  again,  or  from  opposition  to  opposition,  is-  called  a  synodic 
revolution.  Her  passage  from  one  longitude  to  the  same  longitude  again, 
a  tropical  revolution  ;  from  perigee  to  perigee,  or  from  apogee  to  apogee, 
an  anomalistic  revolution  ;  from  one  node  to  the  same  node  again,  a  nodi- 
cal revolution.  The  intervals  of  time  required  to  perform  these  revolutions 
are  called  periods. 

§  449.  To  find  the  length  of  either  of 
these  periods,  say  the  synodic,  let  S  be 
the  sun,  E  the  earth,  M  the  moon  in  con- 
junction, E\  the  place  of  the  earth  at  the 
next  conjunction  of  the  moon,  then  at  M{. 
Draw  JStMi  parallel  to  E  S.  At  the  sec- 
ond conjunction  the  moon  will  have  re 
volved  through  360°  about  the  earth,  in 
creased  by  the  angle  M^E\M\  —  E  S  E± 
=  the  earth's  angular  motion  in  the  same 
time.  Make 

m  =  moon's  mean  daily  motion  ; 
»  =  earth's      " 
t  =  synodic  period. 

Then  tm  =  360°-f-  E  S  E,r 


and  by  subtraction, 


Fig.  84. 


—  tn  =  360°; 


THE    MOON. 

360° 
whence  <  =  -    — •    .    .    .    .         .    .         (146) 

in.  —  -la.  \          t 


m  —  n 


§  450.  Here  n  denotes  the  real  angular  motion  of  the  earth,  which  is 
equal  to  the  apparent  angular  motion  of  the  sun.  If  it  be  replaced  bv 
the  apparent  geocentric  motion  of  the  vernal  equinox,  that  of  the  apogee, 
or  that  of  the  node,  taking  care  to  give  to  each  its  appropriate  sign  (plus 
when  the  motion  is  direct  and  negative  when  retrograde),  the  correspond 
ing  period  will  result.  The  mean  daily  motion  of  the  vernal  equinox  is 
equal  to  50".2  divided  by  365d.242-f  ;  that  of  apogee  to  360°,  divided  by 
the  number  of  mean  solar  days  in  8.854  years;  and  that  of  the  node  by 
the  number  of  days  in  18.6  years. 

The  synodic  period  of  moon  =  29.53  -f-  mean  solar  days. 
The  anomalistic "  "       =  27.55  -f     u  tt 

The  tropical        u  u       =  27.32  -f     "  « 

The  nodical        "  =  27.21  +     u 

The  synodic  period  of  the  moon  is  called  a  lunar  month,  or  lunation. 


Lunar  Phases, 

§  451.  The  sun's  distance  from  the  earth  being  23984,  and  that  of  the 
moon  only  59.96  times  the  earth's  radius,  the  angle  at  the  sun  subtended 
by  the  semi-transverse  axis  of  the  lunar  orbit  is  0°  08'  30";  so  that  rays 
of  light  proceeding  from  the  sun  to  the  moon  and  earth  may  be  regarded 
as  sensibly  parallel;  and  the  exterior  angle  of  elongation  S  P  E',  Fig.  74, 
and  Eq.  (131),  may  be  assumed  equal  to  the  true  elongation  SEP. 
Also  the  variation  in  the  moon's  distance  is  too  small  to  produce  sensible 
change  in  her  apparent  diameter  to  the  naked  eye,  and  the  change  be- 
comes perceptible  only  when  viewed  through  measuring  instrument*. 
The  apparent  diameter  varies  from  29'  21  ".91  to  33'  31".07,  that  at  the 
mean  distance  being  31'  07". 

§  452.  Resuming  Eq.  (131),  making  d  constant,  and  &  equal  to  the 
moon's  elongation,  and  supposing  the  sun  to  the  right  of  the  figure  in  the 
direction  of  E  S  produced,  the  earth  at  JS,  and  the  moon  successively  in 
the  positions  1,  2,  3,  4,  5,  6,  7,  8,  we  shall  find  the  phases  represented  in 
the  figure  on  next  page. 

When  in  conjunction  at  1,  d  or  the  elongation  is  zero,  the  moon  is  in- 
risible,  and  this  phase  is  called  new  moon.  When  at  2,  the  elongation 
being  45°  east,  the  moon  is  said  to  be  in  first  octant,  and  the  phas^  is 


SPHERICAL  ASTRONOMY. 
Fig.  86. 

0) 


crescent.  When  at  3,  the  elongation  being  90°  east,  the  moon  is  said  to 
be  in  first  quarter,  and  the  phase  is  dichotomous.  When  at  4,  the  elon- 
gation being  135°  east,  the  moon  is  said  to  be  in  second  octant,  and  the 
phase  is  gibbous.  When  in  opposition  at  5,  the  elongation  is  180°,  the 
phase  is  full,  and  is  called  full  moon.  When  at  6,  the  elongation  being 
1 35°  west,  the  moon  is  said  to  be  in  third  octant,  and  the  phase  is  gibbous. 
When  at  7,  the  elongation  being  90°  west,  the  moon  is  said  to  be  in  the 
third  quarter,  and  the  phase  is  again  dichotomous.  When  at  8,  the  elou 
Cation  being  45°  west,  the  moon  is  said  to  be  in  fourth  octant^  and  the 
.jhase  is  crescent.  The  interval  of  time  required  for  the  moon  to  pa^s 
through  all  these  phases  and  resume  them  anew,  is  one  synodic  period,  or 
lunation. 

§  453.  The  earth  presents  to  the  moon  the  same  phases  that  the  moon 
does  to  us ;  the  angle  of  elongation  of  the  earth,  as  seen  from  the  moon, 
being  always  the  supplement  of  the  elongation  of  the  moon,  as  seen  from 
the  earth. 

§  454.  The  pale  light  of  the  moon,  by  which  its  outline  is  defined  in 
conjunction,  is  due  to  the  light  reflected  from  the  earth,  then  full,  falling 
upon  the  dark  side  of  the  moon 

ECLIPSES  OF  THE  SUN  AND  MOON. 

§  455.  The  planets  and  sate. lites,  being  opaque,  non-luminous  bodies, 
and  receiving  their  light  from  the  sun,  which  is  of  vastly  greater  size,  cast 
conical  shadows,  of  which  the  surfaces  produced  must  always  be  tangent 
to  the  sun's  surface.  The  axes  of  the  shadows  cast  by  the  planets,  lie  in 
the  planes  of  their  respective  orbits.  That  of  the  earth  is  in  the  plane  of 
the  ecliptic ;  and  if  at  the  time  of  syzygy  the  moon  be  near  one  of  her 
nodes,  she  will  either  pass  within  the  luminous  portion  of  the  conical 


ECLIPSES 


119 


space  between- the  earth  and  sun,  or  enter  the  earth's  shadow,  according 
as  her  phase  is  new  or  full. 

In  the  first  case,  she  will  mask  the  whole  or  part  of  the  sun  from  some 
portions  of  the  earth's  surface ;  and  in  the  latter,  will  suffer  a  loss  of  the 
light  she  herself  receives  from  that  body. 

§  456.  The  obscuration  of  the  sun,  by  the  interposition  of  the  moon 
oetween  the  sun  and  earth,  is  called  a  solar  eclipse.  The  obscuration  of 
the  moon,  by  a  loss  of  solar  illumination  while  within  the  earth's  shadow, 
is  called  a  lunar  eclipse. 

Fig.  86. 


§  457.  Let  S  be  the  sun,  E  the  earth,  D  Fj  and  C  V^  tangents  to  the 
sun  and  earth ;  A  Ft  B  will  be  the  earth's  shadow.  Let  M  be  the  moon 
just  entering  the  shadow,  and  H  H1  a  right  section  of  the  latter  at  the 
distance  of  the  moon.  Make 

if  =  E  C  A   =  sun's  horizontal  parallax  ; 

tf  =  C  E  S   =  sun's  apparent  semi-diameter ; 
P  =  E  H  A  =  moon's  equatorial  horizontal  parallax ; 

s  =  HE  M  =  moon's  apparent  semi-diameter  ; 
R  =  E  A       =  earth's  equatorial  radius ; 

then  in  the  triangle  E  Fi  (7, 

angle  F,  =  tf  —  if ; 

in  the  triangle  E  H  F,, 

angle  H=  180° —  P; 
and  same  triangle, 

E  F,  :  EH  :  :  sin  (180°  —  P)  :  sin  (tf  —  ir); 
whence 

sin  (tf  —  TT)  (f  —  if 

The  least  value  for  P  is  52'  50"  ;  the  greatest  value  for  <f  —  «  is  16'  10"  ; 
whence  the  length  of  the  earth's  shadow  is  always  greater  than  three  times 
the  distance  of  the  moon. 


120 


SPHERICAL    ASTRONOMY. 
Fig.  86  bis. 


458.  Again,  in  same  triangle, 

HE  Vl  =  EH  A  -  EV, 
and  denoting  the  angle  HE  V{  by  E,  we  have 


—  <f  .......     (148) 

The  least  value  for  P  -\-  <K  —  a  is  36'  41  ";  whence  the  apparent  semi- 
diameter  of  the  section  of  the  earth's  shadow  at  the  moon's  distance  is  al- 
ways greater  than  twice  that  of  the  moon  ;  and  this  must  be  increased  by 
about  one-fiftieth  of  its  value  for  the  atmospheric  absorption  of  the  solar 
light  which  passes  near  the  earth's  surface.  The  moon  may,  therefore, 
enter  completely  within  the  earth's  shadow. 

§  459.  Denoting  the  angular  distance  V1EM  between  the  axis  of  the 
earth's  shadow  and  moon's  centre,  at  the  beginning  or  ending  of  the  lunar 
eclipse,  by  ^y,  we  have 

4,  =  E  +  s  =  P  +  *  •  —  ff  +  a      .     .     .     .     (149) 

§  460.  The  conical  space  on  the  opposite  side  of  the  earth  from  the 
sun,  and  of  which  the  bounding  surface  is  tangent  to  these  bodies,  and 
vertex  between  them,  is  called  the  earth's  penumbra.  Thus,  LEAL'  is 
the  earth's  penumbra.  Its  apparent  semi-diameter  L  E  Vy,  at  the  distance 
of  the  moon,  denoted  by  E,,  is  obtained  from  Eq.  (148)  by  simply 
changing  the  sign  of  tf,  these  semi-diameters  falling,  in  this  case,  on 
opposite  sides  of  the  axis  SE;  and  we  have 

Ef  =  P  +  ie  +  (t  .......     (150) 

The  moon  experiences  a  loss  of  light  from  the  instant  she  touches  the  pe- 
numbra, and  this  loss  continues  to  increase  till  she  enters  the  umbra  or 
shadow. 

§  461.  Now,  let  S  be  the  sun,  M  the  moon  at  new,  E  the  centre  of 
the  earth,  B  A  an  arc  of  the  earth's  surface  —  enlarged  to  avoid  confusing 
the  figure.  The  space  N  Vl  Nf  is  the  moon's  shadow,  and  B  N'  N  A  her 
penumbra.  To  all  places  within  the  section  of  the  former  by  the  earth's 
surface,  and  of  which  a  b  is  the  diameter,  the  sun  will  be  totally,  and  to 


121 


all  places  within  the  annular  space,  of  which  B  a  and  b  A  are  sections, 
partially  obscured,  and  present  in  the  latter  case  a  crescent  phase.     Make 

r  =  ME  =.  moon's  distance ; 
r,  =  S  E  =  sun's  distance ; 
d  —  MJV=  moon's  true  semi-diameter; 
d,  =  S  C  =  sun's  true  semi-diameter ; 
x  =  E  F!  =  distance  of  conical  vertex  from  earth's  centre. 

Then,  in  the  triangles  F,  S  C  and   F,  MN,  right-angled  at  C  and  JV, 


whence 


r  —  x      r/  —  x 


x  =. 


d,r-dr, 


d,-d 

a'-a  substituting  the  values  of  r,  r,,  d,  and  c?x,  as  given  by  equations  (28) 

and  (29), 

rf  —  a 

(181) 


—  xs 


§  462.  Taking  the  values  for  P,  <r,  a1,  and  s,  which  give  this  the  greatest 
positive  and  negative  values,  it  is  found  that  the  vertex  F,  sometimes  falls 
short  of  the  earth's  centre  about  7.6,  and  at  others  extends  beyond  that 
point  about  3.5  times  the  earth's  radius. 

§  463.  Again,  R  —  x  gives  the  distance  of  the  vertex  F1}  from  the  sec- 
tion of  which  a  &  is  the  diameter.  Denoting  this  latter  by  y,  we  have,  in 
the  triangles  a  F  6  and  N  F,  N', 

r  —  x  :  R  —  x  :  :  2d  :  y: 


r  —  x 


(152) 


and  substituting  the  values  of  c?,  a?,  and  r,  from  equations  (29),  (151),  and 
(28),  we  find 

l;±9    .   .   .  .053) 


122  SPHERICAL    ASTRONOMY". 

§  464.  As  long  as  R  —  x  is  positive,  y,  Eq.  (152),  will  be  positive,  and 
the  shadow  will  reach  and  cut  the  earth.  To  all  places  within  the  boun- 
dary of  this  intersection,  the  sun  will  be  wholly  invisible,  and  the  eclipse 
is  said  to  be  total.  The  greatest  value  for  y  positive  is  about  170  miles. 

§  465.  When  R  —  x  is  zero,  the  vertex  of  the  shadow  just  comes  tc 
the  earth,  and  the  sun  can  be  totally  eclipsed  only  to  one  place  at  a  time. 

§  466.  When  R  —  x  is  negative,  the  vertex  falls  short  of  the  earth,  and 
the  surface  of  the  latter  intersects  the  opposite  nappe  of  the  conical  shadow; 

Fig.  88. 


the  value  of  y  is,  Eq.  (152),  negative;  it  measures  the  distance  by  which 
the  opposite  edges  of  the  inner  boundary  of  the  penumbra  overlap  one  an- 
other, and  the  space  of  which  y  negative  is  the  diameter  may  be  called  the 
umbral  penumbra,  and  is  distinguished  from  the  rest  of  the  penumbra  in 
embracing  those  points  from  which  the  sun  appears  as  an  unbroken  ring 
around  the  black  disk  of  the  moon,  while  to  all  other  points  of  the  penum- 
bra he  will  appear  as  crescent.  In  the  first  case  the  eclipse  is  said  to  be 
annular  ;  in  the  second,  crescent.  The  greatest  possible  diameter  of  the 
umbral  penumbra  is  about  240  miles. 

§  467.  To  find  AB,  the  diameter  of  the  external  boundary  of  the  pe- 
numbra on  the  earth,  it  is  only  necessary  to  change  the  sign  of  s  in  Eq. 
(153),  because  in  this  case  <f  and  s  fall  on  opposite  sides  of  the  axis  of  the 
moon's  shadow.  Making  this  change  in  Eq.  (153),  we  have 


The  greatest  value  for  which  is  about  4835  miles. 

§  468.  The  solar  eclipse  begins  at  the  instant  of  first,  and  ends  at  the 
instant  of  last  contact  of  the  moon  with  the  cone  tangent  to  the  sun  and 
earth  ;  and  the  places  of  first  and  last  appearance  on  the  earth  are  those  at 
which  the  corresponding  rectilinear  elements  of  this  cone  are  tangent  to  its 
surface.  The  solar  eclipse  being  only  visible  to  those  places  situated  with- 
in the  path  of  the  penumbra,  is,  when  considered  with  reference  to  the 


ECLIPSES. 

whole  earth,  called  a  general  eclipse  of  the  sun,  in  contradistinction  to  its 
local  character,  of  which  more  will  be  said  presently. 

§  469.  Let  MI  (Fig.  86)  be  the  place  of  the  moon  at  the  beginning  of  a 
general  eclipse  of  the  sun  ;  denote  her  angular  distance  M%  E  S  from  the  sun 
by  4  ;  then,  since  EKB  =  P,  E  Vl  D  =  <f  —  *,  and  M2E  K  =  s,  will 


(155) 


§  470.  In  a  lunar  eclipse,  if  the  moon's  centre  should  cross  the  axis  of 
the  earth's  shadow  the  eclipse  is  said  to  be  central.  When,  during  a  solar 
eclipse,  the  centre  of  the  sun,  that  of  the  moon,  and  the  eye  of  the  specta- 
tor are  on  the  same  right  line,  the  eclipse  is  said  to  be  central.  If  at  time 
of  syzygy  the  moon  become  tangent  to  the  cone  of  the  earth's  shadow 
without  entering,  the  phenomenon  is  called  appulse. 

§  471.  The  atmospheric  lens  which  envelops  the  earth  causes  the  solar 
light  passing  through  it  to  converge  to  a  focus  between  the  moon  and 
earth,  and  this  light  diverging  anew  after  concentration,  and  falling  upon 
the  lunar  disk  while  in  the  earth's  shadow,  gives  to  it  a  dark,  coppery-red 
illumination,  and  prevents  total  obscuration  of  the  moon  during  a  lunar 
eclipse. 

Relative  Geocentric  Orbit  of  the  Moon. 

§  472.  The  path  which  a  body  in  motion  appears  to  describe  in  refer- 
ence to  another  also  in  motion,  is  called  a  relative  orbit  ;  and  the  distance 
of  the  one  body  from  the  other  at  any  time  will  be  the  same  whether  we 
regard  both  as  moving  with  their  actual  velocities,  or  one  at  rest  and  the 
other  moving  with  a  velocity  of  which  the  components  in  any  three  rec- 
tangular directions  are  equal  to  the  differences  of  the  components  of  the 
actual  velocities  in  the  same  directions. 

§  473.  Let  NO  be  an  arc  of  the 
ecliptic,  N  L  an  arc  of  the  lunar  orbit 
projected  upon  the  celestial  sphere,  N 
one  of  the  nodes,  S  the  point  in  which 
the  axis  of  the  earth's  shadow  pierces  the 
celestial  sphere  when  the  moon  is  in  her 

node,  0  the  place  of  this  point  when  the  moon  is  either  in  conjunction  or 
opposition  at  L.  From  the  node  to  opposition  or  conjunction  the  moon 
will  have  described  the  arc  N  L  and  the  axis  of  the  earth's  shadow,  whose 
motion  is  always  equal  to  the  apparent  motion  of  the  sun,  the  arc  S  0. 
L  0  is  an  arc  of  a  circle  of  latitude  ;  and  assuming  NL  to  represent  the 


SPHERICAL    ASTRONOMY. 

moon's  actual  velocity,  N  0  and    0  L  Fi£-  S9  b!#- 

will  be  its  components  in  longitude  and 

latitude  respectively.    S  0  is  the  velocity 

of  the  earth's  shadow,  which  is  wholly  in 

longitude.      If,  therefore,   we   construct 

upon  ATS  =  ArO-SO,   audSM  = 

//  0,  the  rectangle  S  JB,  the  diagonal  JV^J^will  be  an  arc  of  the  relative 

orbit  of  the  moon  referred  to  the  axis  of  the  earth's  shadow  as  an  origin. 

Ecliptic  Limits. 

§  474.  Hence,  if  S  M}  be  drawn  perpendicular  to  the  relative  orbit, 
the  length  of  S  MI  will  measure  the  nearest  approach  of  the  moon's  cen- 
tre to  the  axis  of  the  earth's  shadow.  Make 

m  =  moon's  hourly  motion  in  longitude  ; 
n  —  sun's         "  "  " 

g  =  moon's  hourly  motion  in  latitude  ; 
t  =  time  from  node  to  opposition  or  conjunction  ; 
<p  =  M  N  S  =  angle  which  the  relative  orbit  makes  with  the  ecliptic. 

MS 

tan9  =  —  ; 

but  supposing  the  motion  uniform, 


also 

N0  =  mt;  S  0  =  nt', 
and 

.-.  JV  S  =  m  t  —  n  t  —  (m  —  n)  t  ; 
whence 


tan  9  =  —   —      .....     •     .     (156) 

m—  -  n 


The  angle  <p  is  then  known. 

§  475.  Denoting  S  Ml  by  J,  we  have 


SJV=-^- -  =  4^1 "'  ^y    .     .     .    .     (157) 

sin  9  g 

Also 


m 


m- 
whence,  substituting  the  value  of  S  JV, 


_?!-...  yv"*-'*;  i-y      B    ^    (158) 
m  —n  a 


ECLIPSES. 


125 


Making  d  equal  to  that  given  in  Eq.  (149),  the  moon  will  just  touch  the 
earth's  shadow,  and  N  0  will  become  what  is  called  the  ecliptic  limit  ; 
that  is,  the  least  difference  of  longitude  that  can  exist  between  the  moon, 
and  Jier  nearest  node  at  full,  to  avoid  an  eclipse  of  the  moon. 

Taking  the  greatest  value  for  A,  NO  is  found  to  be  12°  24',  and  least 
value  it  is  found  to  be  9°.  The  first  is  called  the  greatest  and  the  second 
the  least  lunar  ecliptic  limit.  If  therefore  at  the  time  of  full  moon  the 
difference  between  the  longitude  of  the  moon  and  her  nearest  node  exceed 
12°  24',  there  cannot  be  an  eclipse;  if  less  than  9°,  there  must  be  one; 
if  less  than  12°  24'  and. greater  than  9°,  there  may  or  may  not,  depend- 
ing upon  the  inclination  of  the  relative  orbit  and  actual  value  of  d.  To 
solve  the  doubt,  we  have  the  given  difference  of  longitude  between  12°  24' 
find  9°,  and  the  inclination  <p,  to  find  S  Mv  If  this  latter  be  greater  than 
^/,  there  can  be  no  eclipse ;  if  less,  there  must  be  one. 

§  476.  Again,  making  d  equal  to  that  given  in  Eq.  (155),  and  pro- 
ceeding exactly  as  above,  we  find  the  greater  and  lesser  solar  ecliptic  lim- 
it*. The  first  is  18°  36'  and  the  latter  15°  25'. 

Number  of  Eclipses. 

g  477.  LetNHN'H'  be  the  ecliptic, 
N  and  JV'  the  moon's  nodes.  Take 
NL{,NLZ,  AP.L3,and  N'Lt,  each  equal 
to  1 8°.6,  the  greatest  ecliptic  limit.  Then 
will  LI  Z,  and  L3  L4  be  each  equal  to 
37°.2,  and  the  number  of  new  moons 
that  can  happen  while  the  sun  is  appa- 
rently describing  these  arcs,  will  deter- 
mine the  number  of  solar  eclipses  that 
can  occur  in  a  single  year. 

The  mean  daily  motion  of  the  moon's 
node  is  —  0°.055  ;  the  mean  apparent  daily  motion  of  the  sun  is  0°.985, 
and  hence  the  apparent  relative  motion  of  the  sun  and  node  is  0°.985  — 
(—  0°.055)  =  1°.04.  A  lunation  is  29.53  days;  and  29.53  X  1°.04  = 
30°/7112,  say  30°. 71,.  is  the  mean  motion  of  the  sun  from  the  node  in  a 
lunation.  Omitting  the  proper  motion  of  the  vernal  equinox  as  insignifi- 
cant in  this  estimate,  its  relative  motion  from  the  node  in  a  lunation  is 
29.53  days  x  0°.055  =  1°.6241.  These  motions  are  incommensurable 
with  each  other  and  with  360° ;  and  the  vernal  equinox,  the  node  and 
sun  with  the  moon  in  conjunction,  will,  in  process  of  time,  have  any  as- 
sumed positions  with  respect  to  each  other  at  the  beginning  of  the  year. 


126  SPHERICAL   ASTRONOMY. 

Taking  the  sun  at  M^  one  degree  to  the  east  of  Ll  (with  moon  in  con- 
junction), will  give  one  solar  eclipse;  and  37°. 2  —  1°=:  36°. 2  being 
greater  than  the  arc  described  by  the  sun  in  a  lunation,  there  will,  at  tht 
end  of  the  first  lunation,  be  another  solar  eclipse  between  N  and  Lz. 
At  the  end  of  the  sixth  lunation,  the  sun  will  be  at  M^  in  advance  of 
L3  by  a  distance  equal  to  30°.7l  X  6  —  179°  =  5°.26,  where  there  will 
be  a  third  solar  eclipse ;  and  37°. 2  —  5°. 26  =  31°. 84  being  greater  thau 
arc  described  in  a  lunation,  there  must  be  a  fourth  solar  eclipse  before 
the  sun  passes  L4.  At  the  end  of  the  twelfth  lunation,  the  sun  will  be 
30°.7l  x  12  —  360°  =  8°. 52  to  the  east  of  J/~g,  his  initial  place,  where 
there  will  be  a  fifth  solar  eclipse,  and  this  will  be  the  last  within  the 
year,  which  will  end  10.89  days  after,  this  being  the  excess  of  the  year 
over  twelve  lunations. 

Again,  18°. 6  —  1°=  17°. 6;  and  as  in  ^a  semi-lunation  the  sun  will 
pass  over  15°. 35  of  this  arc,  he  will  come  to  the  distance  17°. 6  — 
15°.35  =  2°. 25  from  the  node  N^  and  there  will  be  a  first  lunar  eclipse 
at  the  opposite  node  JV^.  When  the  moon  was  new  at  M^  the  sun  was 
18°. 60  —  5°. 36  =  13°.34  from  the  node  N,  and  in  half  a  lunation  after 
will  be  15°.35  —  13°.34  =  2°.01  beyond  it,  and  there  will  be  .1  second 
lunar  eclipse,  and  no  more  within  the  year,  for  the  next  lunation  will 
carry  the  sun  beyond  the  lunar  ecliptic  limits.  Had  the  initial  place  of 
the  sun  been  taken  at  M^  at  a  distance  4°. 26  to  the  west  of  the  node 
N,  at  the  beginning  of  the  year,  and  the  moon  in  opposition,  it  might 
have  been  shown  that  there  would  have  been  four  eclipses  of  the  sun 
and  three  of  the  moon. 

§  478.  The  least  solar  ecliptic  limit  being  15°.42,  the  arc  L^  L^  must 
be  at  least  30°. 84 ;  which  being  greater  than  the  arc  passed  over  by  the 
sun  in  a  lunation,  there  must  be  at  least  one  solar  eclipse  in  each  of  the 
arcs  Ll  L^  and  L^  L^  so  that  there  must  always  be  at  least  two  eclipses 
of  the  sun  in  each  year. 

The  sun  is  less  than  a  lunation  in  passing  through  the  lunar  ecliptic 
limits,  and  there  may,  therefore,  be  no  eclipse  of  the  moon  within  the  year. 

To  sum  up,  then,  there  may  be  seven  eclipses  within  the  year,  and  theie 
may  be  only  two.  In  the  former  case,  five  may  be  of  the  sun  and  two  of  the 
moon,  or  four  of  the  sun  and  three  of  the  moon ;  and  in  the  latter,  both 

must  be  of  the  sun. 

The  Saros. 

§  479.  The  synodic  period  of  the  moon  is  29.53058,  and  that  of  the 
moon's  node  "346.6196  days.  These  numbers  are  to  one  another  as  19  to 
223  nearly.  If,  therefore/ the  moon  and  her  nodes  be  in  syzv-ry'at  tho 


Plate  ^L 


TO  FRONT  JSAGE 


CONSTITUTION    OF    THE    MOON. 


127 


same  time,  they  will  be  so  again  after  19  revolutions  of  the  node,  or  223 
lunations ;  so  that  the  eclipses  will  recur  again  very  nearly  in  the  same 
order  within  the  same  period,  which  is  about  18.027  years.  This  period 
is  known  as  the  Chaldean  Saros.  There  are  generally  70  eclipses  in  the 
saros,  of  which  29  are  lunar  and  41  solar. 


PHYSICAL  CONSTITUTION  OF  THE  MOON. 

§  480.  Telescopes  disclose  certain  varieties  of  illumination  Oi  the 
moon's  surface,  which  can  only  arise  from  mountains  and  valleys.  The 
shadows  cast  by  the  former  lie  in  directions  and  are  of  lengths  re- 
quired by  the  inclination  of  the  solar  rays  to  that  portion  of  the  moon's 
surface  on  which  the  mountains  stand.  The  convex  outline  of  the  moon 
turned  towards  the  sun  is  always  circular  and  nearly  smooth ;  but  the  op- 
posite or  elliptical  border  of  the  illuminated  part  is  extremely  ragged,  and 
indented  with  deep  recesses  and  prominent  points.  To  places  along  this 
line  the  sun  is  just  rising,  and  the  neighboring  mountains  cast  long  black 
shadows  on  the  plains  below.  As  the  sun  rises  these  shadows  shortei, ; 
and  at  full  moon,  when  the  solar  light  penetrates  the  mountain  valleys  and 
shines  on  every  point  of  the  field  of  view,  no  shadows  are  seen. 

§  481.  The  summits  of  the  lunar  mountains  often  appear  as  small 
Bright  points,  or  islands  of  light,  beyond  the  edge  of  the  illuminated  part, 
as  they  catch  the  sunbeams  before  the  intervening  plains.  As  the  sun  ad- 
vances in  altitude,  these  luminous  patches  expand,  and  finally  unite  with 
the  general  illumination,  and  the  mountains  appear  as  projections  from  its 
elliptical  border. 

§  482.  To  compute  the 
height  of  a  lunar  mountain, 
let  E,  M,  and  S  be  the  cen- 
tres of  the  earth,  moon,  and 
sun  respectively  ;  AC B  D 
and  DOC  sections  of  the 
general  surface  of  the  moon 
by  planes  respectively  perpen- 
dicular to  EM  and  MS] 
then  will  the  visible  illumi- 
nated part  of  the  disk  be 

contained  between  CBD  and  the  projection  of  D  0  C  on  the  section 
AC  B  D.  Also  let  m  be  the  top  of  a  mountain  just  catching  the  solar 
rays  that  graze  th*}  general  surface  of  the  moon  at  0 ;  EOF  the  arc  of  a 


128 


SPHERICAL   ASTRONOMY. 


great  circle  of  this  surface, 
and  of  which  the  plane  passes 
through  the  top  of  the  moun- 
tain and  centres  of  the  sun 
and  moon ;  n  the  point  in 
which  this  arc  is  cut  by 
the  line  J\fm  drawn  from 
the  top  of  the  mountain  to 
the  moon's  centre. 

Make 

r  —  M  n      =  radius  of  the  moon ; 

s  =  FC  0  =  E  M A  =  exterior  angle  of  elongation  ; 

y  =  0  m     =  distance  of  m  from  0 ; 

x  =  m  n      =  height  of  mountain ; 

a  —  the  projection  of  y  on  the  plane  AGED. 

Then,  since  the  ray  S'  0  m  is  perpendicular  to  the  section  D  0  (7,  it  is  in- 
clined to  the  section  A  C  B  D,  under  an  angle  equal  to  the  complement 
of  F  C  0  =  90°  —  e ;  and  we  have 

a  =  y  .  cos  (90°  —  s)  =  y  .  sin  s ; 

whence 

a 


y  = 


sin  s 


Also 


and,  therefore, 


a 

sin  s 


neglecting  x  in  comparison  with  2  r,  we  have 
a2        1 


x  = 


—  .  T-S-  =  —  •  cosec2 


2  r  '  sin 


2r 


(159) 


a  being  the  observed  distance  of  the  bright  spot  from  the  boundary  of  the 
illumination,  may  be  measured  by  means  of  the  micrometer. 

§  483.  The  heights  of  many  of  the  lunar  mountains  have  been  thus 
computed,  and  they  range  through  all  elevations  up  to  23,000  English  feet. 

§  484.  The  lunar  mountains  are  strikingly  uniform  in  aspect.  They 
are  very  numerous,  especially  towards  the  southern  border,  occupying  by 
far  the  larger  portion  of  the  surface.  They  present  almost  universally  a 
circular  or  cup-shaped  form  in  ground  plan,  which  becomes  foreshortened 
into  an  ellipse  towards  the  limb.  The  larger  of  these  cups  have  for  the  most 


CONSTITUTION    OF    THE    MOON. 


129 


part  flat  bottoms,  from  each  of  which  rises  centrally  a  small,  steep,  conical 
hill,  presenting  in  all  respects  the  true  volcanic  character  as  exhibited  by 

Fig.  92. 


like  districts  on  the  earth,  but  with  this  peculiarity,  viz. :  that  the  bottoms 
are  so  deep  as  to  lie  below  the  general  surface  of  the  moon,  the  internal 
depth  being  often  twice  or  thrice  the  external  height. 

§  485.  The  heights  of  mountains  in  the  immediate  vicinity  of  each 
other  being  proportional  to  the  length  of  their  respective  shadows,  the 
depths  of  the  pits  or  craters  are  easily  computed  from  the  heights  of  the 
edges  above  the  general  level,  and  the  lengths  of  the  shadows  they  cast 
internally  and  externally. 

§  486.  Through  the  Rosse  telescope,  the  flat  bottom  of  the  crater 
called  AlbategniuSj  is  seen  to  be  strewed  with' blocks  not  visible  through 
inferior  instruments ;  and  the  exterior  of  another,  called  Aristillus,  is 
hatched  over  with  deep  gullies,  radiating  from  a  centre. 

§  487.  There  are  also  extensive  tracts  of  the  lunar  surface  which  are 
perfectly  level,  and  present  decided  indications  of  an  alluvial  characterr 
and  yet  there  is  a  total  absence  of  all  appearances  of  deep  water. 

§  488.  There  are  no  clouds,  or  other  indications  of  an  atmosphere. 

A  lunar  atmosphere  of  a  mean  density  equal  to  1980th  that  of  the 
earth,  would  give  a  horizontal  refraction  of  1",  and  cause  the  diameter  of 
the  moon,  measured  with  a  micrometer  and  estimated  by  the  interval  of 
a  star's  disappearance  in  an  occultation,  to  differ ;  would  cause  the  limb  of 
the  moon,  during  a  solar  eclipse,  to  appear  beyond  the  cusps  externally  to 
the  sun's  disk  as  a  narrow  line  of  light,  extending  for  some  distance  along 
the  edge ;  and  would  extinguish  very  faint  stars  before  occultations.  But 
none  of  these  phenomena  are  seen.  During  the  continuance  of  a  total 
lunar  eclipse,  when  the  light  of  the  moon  is  so  deadened  as  not  to  obliter- 
ate by  contrast  the  feeble  light  of  the  smaller  stars,  the  latter  are  seen  to 
come  up  to  the  moon's  limb  and  undergo  sudden  extinction,  without  any 
apparent  displacement. 

§  489.  The  light  from  the  moon  developes  but  feeble  lieat,  for  even 

9 


130  SPHERICAL    ASTRONOMY. 

when  collected  into  the  foci  of  large  reflectors,  it  affects  but  little  the 
thermometer ;  and  there  are  no  appearances  indicating  the  slightest 
change  of  surface,  such  as  would  result  from  the  periodical  growth  and 
decay  of  vegetation  which  accompany  a  change  of  seasons. 

§  490.  To  an  inhabitant  of  the  moon,  if  there  be  such  a  thing,  the 
earth  must  present  the  appearance  of  a  moon  2°  in  diameter,  exhibiting 
phases  complementary  to  those  the  moon  presents  to  us,  but  fixed  in  the 
sky,  while  the  stars  seem  to  pass  slowly  beside  and  behind  it.  It  must 
appear  clouded  with  variable  spots,  and  belted  with  zones  corresponding  to 
our  trade-winds.  During  a  solar  eclipse  our  atmosphere  will  appear  as  a 
narrow,  bright  ring,  of  a  ruddy  color  where  it  rests  on  the  earth,  gradually 
passing  into  faint  blue,  encircling  the  whole  or  part  of  the  earth's  disk. 


SATELLITES  OF  JUPITER. 

§  491.  The  satellites  of  Jupiter,  four  in  number,  revolve  about  their  pri- 
mary from  west  to  east  in  planes  nearly  coincident  with  that  of  the  planet's 
equator,  and  but  slightly  inclined  to  the  ecliptic. 

§  492.  Their  orbits  appear,  therefore,  projected  very  nearly  into  straight 
lines,  in  which  they  oscillate  to  and  fro,  sometimes  passing  between  the 
sun  and  Jupiter,  causing  an  eclipse  of  the  sun  to  the  latter,  sometimes  en- 
tering the  planet's  shadow  and  being  themselves  eclipsed,  and  sometimes 
disappearing  either  behind  the  body  of  Jupiter  or  in  transiting  his  disk. 

§  493.  Thus,  let  S  be  the  sun;  E,  the  earth,  of  which  the  orbit  i& 
EFGH-,  J,  Jupiter ;  and  efa  6,  the  orbit  of  a  satellite.  The  cone  of  Ju- 
piter's shadow  will  have  its  vertex  at  Jf,  far  beyond  the  orbit  of  the  satel- 

Fig.  93. 


SATELLITES    OF    JUPITER.  13} 

lite,  and  the  penumbra,  owing  to  the  great  distance  of  the  sun  and  conse- 
quent smallness  of  the  angle  at  Jupiter  subtended  by  his  disk,  will  extend 
but  little  beyond  the  shadow  within  the  limits  of  the  satellite's  orbit. 
The  satellite  revolving  from  west  to  east,  will  cast  a  shadow  upon  Jupiter 
while  passing  from  in  to  w,  will  transit  his  disk  from  e  to  /,  enter  his 
shadow  at  a,  emerge  from  it  at  6,  and  disappear  behind  the  body  of  the 
planet  while  passing  from  c  to  d. 

§  494.  The  shadows  of  the  satellites  are  frequently  seen  crossing  the 
disk  of  Jupiter.  While  in  the  act  of  transiting,  the  satellite  generally  dis- 
appears, its  light  being  confounded  with  that  of  the  planet,  unless  it  hap- 
pens to  be  projected  upon  a  dark  belt,  in  which  case  it  is  visible.  Under 
these  circumstances  it  occasionally  appears  as  a  dark  spot  smaller  than  its 
shadow,  which  has  led  to  the  conclusion  that  certain  of  the  satellites  have 
now  and  then  on  their  own  bodies,  or  within  their  atmospheres,  obscure 
spots  of  great  extent. 

§  495.  From  the  eclipses  of  the  satellites  are  obtained  all  the  data  for 
the  determination  of  the  laws  of  their  motions.  These  eclipses  are  in  gen- 
eral analogous  to  those  of  the  moon,  but  in  their  details  they  differ  con- 
siderably. The  great  distance  of  Jupiter  from  the  sun  and  his  great  size, 
make  his  shadow  much  larger  and  longer  than  that  of  the  earth.  The  sat- 
ellites are  much  smaller  in  proportion  to  their  primary,  and  their  orbits  less 
inclined  to  his  ecliptic,  than  in  the  case  of  the  moon.  From  these  causes 
the  three  interior  satellites  enter  the  shadow  at  every  revolution,  and  are 
totally  eclipsed ;  and  although  the  fourth,  from  the  greater  inclination  and 
distance  of  its  orbit,  sometimes  escapes  eclipse,  yet  it  does  so  seldom. 

§  496.  Besides,  these  eclipses  are  not  seen  by  us  from  the  centre  of  mo- 
tion, as  are  those  of  the  moon,  but  from  some  remote  station,  of  which  the 
place  with  respect  to  the  shadow  is  ever  changing.  And  while  this  cir- 
cumstance makes  no  difference  in  the  time  of  the  eclipses,  it  yet  affects 
materially  the  visibility  and  the  apparent  relative  situations  of  the  planet 
and  satellites  at  the  instant  of  the  latter's  entering  and  quitting  the  shadow 

'§  497.  A  satellite  never  enters  the  shadow  suddenly  because  of  its  sen 
sible  diameter,  and  the  time  from  the  first  perceptible  loss  of  light  to  its 
total  extinction  will  be  that  required  by  the  satellite  to  describe  about  Ju- 
piter an  angle  equal  to  its  apparent  diameter  as  seen  from  the  planet's  cen- 
tre. The  same  is  true  of  the  emergence.  Owing  to  the  difference  in  tel- 
escopes and  eyes,  this  becomes  a  source  of  discrepancy  in  the  times  assigned 
by  different  observers  for  the  beginning  and  ending  of  an  eclipse.  But  il 
both  the  immersion  and  emersion  be  observed  by  the  same  person  and  with 
the  same  telescope,  the  half  sum  of  the  two  times,  as  given  by  a  property 


132 


SPHERICAL    ASTRONOMY. 


regulated  time-keeper,  will  be  that  of  apparent  opposition  measurably  fre« 
from  error. 

§  498.  The  intervals  between  the  oppositions  give  the  synodic  period, 
which,  in  Eq.  (146),  will  give  the  mean  motion,  knowing  that  of  Jupiter, 
and  hence  the  sidereal  period.  Eq.  (142). 

The  satellites  are  named  first,  second,  third,  and  fourth,  according  to 
their  order  of  distance  from  Jupiter. 

The  elements  of , the  satellites'  orbits  will  be  found  in  the  following 

Table. 


Sat 
1st. 

Sidereal  period. 

Inclination  of 
Mean           orbit  to  a 
distance.       fixed  plane 

Rad.ofJ=l.Pr°Pertoeach 

Inclination 
of  the 
fixed  plane  to 
Jupiter's 
equator. 

Retrograde 
revolution  of 
nodes  on 
fixed  plane. 

Mass: 
that  of  Jupiter 

1,000,000,000. 

d.    h.    rn.    s. 
1  18  27  33.506 

.      0       '        " 

6.04853  !    0     0     0 

006 

Years. 

17328 

2d. 

3  13  14  36.393 

9.62347  !    0  27  50 

0     1     5 

29.9142 

23235 

3d. 

7  03  42  33.362 

15.35024      0  12  20 

052 

141.7390 

88497 

4th. 

16  16  31  49.702 

26.99835      0  14  58 

0  24     4 

531.0000 

42659 

It  will  assist  in  forming  some  idea  of  the  relative  dimensions  of  Jupitei 
and  his  satellites  to  examine  the  following 

Table. 


Mean  apparent 
diameter  as  seen 
from  earth. 

Mean  apparent 
diameter  as  seen 
from  Jupiter. 

Diameter 
in 
miles. 

Maes. 

Jupiter. 

38.327 

/         n 

87000 

1.0000000 

1st  sat. 

1.017 

33  11 

2508 

0.0000173 

2d     " 

0.911 

17  35 

2068 

0.0000232 

3d     « 

1.488 

18  00 

3377 

0.0000885 

4th  " 

1.273 

8  46 

2890 

0.0000427 

Prom  which  it  follows  that  the  first  satellite  appears  to  a  spectator  ou 
Jupiter  as  large  as  our  moon  to  us ;  the  second  and  third  nearly  equal  to 
each  other,  and  somewhat  more  than  half  the  size  of  the  first ;  and  the  fourth 
about  a  quarter  of  that  size.  They  frequently  eclipse  each  other.  The 
apparent  diameters  of  the  planet  as  seen  from  the  satellites  are  19°  49'; 
12°  29';  7°  47';  4°  25'. 

§  499.  Figure  93  shows  that  the  eclipses  take  place  to  the  west  of  Ju- 
piter, while  the  latter  is  moving  from  conjunction  to  opposition,  and  to  the 


SATELLITES    OF    JUPITER.  133 

east  from  opposition  to  conjunction.  As  Jupiter  approaches  to  oppo- 
sition, the  line  of  sight  from  the  earth  becomes  more  nearly  coincident 
with  the  direction  of  the  shadow,  and  the  place  of  the  eclipse  will  be 
nearer  and  nearer  to  the  body  of  the  planet.  When  the  earth  comes  to 
F,  from  which  a  line  drawn  tangent  to  the  body  of  the  planet  will  pass 
through  &,  the  emersion  will  cease  to  be  visible,  and  will,  up  to  the  time 
of  opposition,  take  place  behind  the  planet.  Similarly,  from  opposition 
up  to  the  time  when  the  earth  arrives  at  K,  the  immersion  will  be  con- 
cealed from  view.  These  remarks  apply  particularly  to  the  third  and 
fourth  satellites,  the  proximity  of  the  others  to  the  planet  being  so  great 
as  to  make  it  impossible  ever  to  see  the  immersion  and  emersion  both  at 
the  same  eclipse. 

§  500.  The  mean  motions  of  the  satellites  are  connected  by  this  re- 
markable law,  viz :  If  the  mean  angular,  velocity  of  the  first  satellite  be 
added  to  twice  that  of  the  third,  the  sum  will  equal  three  times  that 
of  the  second.  If,  therefore,  from  the  mean  longitude  of  the  first  satel- 
lite, increased  by  twice  that  of  the  third,  three  times  the  mean  longitude 
of  the  second  be  subtracted,  the  remainder  will  be  a  constant  quantity 
and  this  constant  is  found  to  be  equal  to  180°.  This  Laplace  has  shown 
to  be  a  consequence  of  the  mutual  attractions  of  the  satellites  for  one 
another.  The  first  three  satellites  cannot,  therefore,  be  eclipsed  at  the 
same  time. 

§  501.  While,  however,  the  satellites  cannot  all  be  eclipsed  at  once, 
they  may  be,  and,  indeed,  occasionally  arc,  all  invisible  by  the  simulta- 
neous eclipse  of  some,  occupations  of  others,  and  transits  of  the  rest. 

§  502.  The  orbits  of  the  satellites  are  but  slightly  eccentric,  the  two 
inferior  ones  not  at  all  so,  so  far  as  observation  is  capable  of  revealing 
eccentricity.  Their  mutual  attractions  produce  in  them  perturbations 
analogous  to  those  of  the  planets  about  the  sun.  These  are  investigated 
in  physical  astronomy. 

§  503.  By  careful  observations  the  satellites  are  found  to  exhibit 
marked  fluctuations  in  respect  to  brightness.  These  fluctuations  happen 
periodically,  and  appear  connected  with  the  position  of  the  satellites 
with  respect  to  the  sun;  from  which  it  is  inferred  that  they  revolve 
upon  their  axes  like  our  moon,  each  once  in  its  sidereal  period. 

§  504.  At  one  time  the  eclipses  of  Jupiter's  satellites  were  much  used 
i.i  the  determination  of  terrestrial  longitude,  but  more  modern  methods, 
free  from  the  objections  referred  to  in  §  497,  have  in  a  measure  sup- 
planted them. 


134  SPHERICAL    ASTRONOMY. 

Progressive  Motion  of  Light. 

§  505.  1  /  these  eclipses  science  is  indebted  for  the  discovery  of  the  suc- 
cessive propagation  and  velocity  of  light. 

The  earth's  orbit  being  concentric  with  that  of  Jupiter  and  interior  to  it, 
the  distance  of  these  bodies  is  continually  varying,  the  variation  extending 
from  the  sum  to  the  difference  of  the  radii  of  the  two  orbits,  making  the 
excess  of  the  greatest  over  the  least  distance  equal  to  the  diameter  of  the 
earth's  orbit.  Now,  it  was  observed  by  Roemer,  a  Danish  astronomer,  on 
comparing  together  the  eclipses  during  many  successive  years,  that  those 
which  took  place  about  opposition  were  observed  earlier,  and  those  about 
conjunction  later  than  an  average  or  mean  time  of  occurrence.  And  con- 
necting the  observed  acceleration  in  the  one  case  and  retardation  in  the 
other  with  the  variation  of  Jupiter's  distance  below  and  above  its  average 
value,  he  found  the  difference  fully  and  accurately  accounted  for  by  allow- 
ing 16m  268.6  for  light  to  traverse  the  diameter  of  the  earth's  orbit.  In 
other  words,  using  the  figure  of  a  cord  moving  in  the  direction  of  its  length 
from  the  satellite  to  the  earth  to  illustrate  the  flow  of  luminous  waves  in 
the  same  direction,  if  the  cord  were  severed  at  the  edge  of  Jupiter's  shadow, 
the  severed  end  would  be  16™  268.6  longer  in  reaching  the  earth  when  the 
planet  is  in  conjunction  than  in  opposition,  having  a  greater  distance  to 
travel  in  the  first  case  by  the  diameter  of  the  earth's  orbit  =  190,000,000 
miles,  than  in  the  second.  The  satellite  is  seen  long  after  it  has  entered 
the  shadow,  and  is  invisible  long  after  it  has  emerged  from  it.  Dividing 
the  diameter  of  the  earth's  orbit  by  16m  2  6s.  6  reduced  to  seconds,  the  ve- 
locity of  light  is  found  to  be  192,000  miles  a  second. 

SATELLITES  OF  SATURN. 

§  506.  Eight  satellites  are  known  to  accompany  Saturn.  They  revolve 
about  him  from  west  to  east,  and  in  planes  nearly  coincident  with  that  of 
the  planet's  ring,  except  the  eighth,  whose  orbit  is  inclined  to  this  latter 
plane  under  an  angle  of  about  12°  14'.  This  satellite  is  also  distinguished 
from  the  others  by  its  remoteness  from  the  planet,  its  distance  being 
2.3  times  that  of  the  most  distant  of  the  others,  and  equal  to  64  times 
the  equatorial  radius  of  Saturn,  resembling  in  this  respect  our  own  moon. 
It  is  also  remarkable  for  the  exhibition  of  greater  variety  of  illumination 
in  different  parts  of  its  orbit  than  any  other  known  secondary.  Indeed,  so 
feeble  is  the  light  which  it  reflects  to  the  earth  when  to  the  east  of  Saturn 
that  it  becomes  invisible  through  ordinary  telescopes  ;  and  from  this  deti- 


SATELLITES    OF   SATURN. 


135 


ciency  of  light  occurring  constantly  on  the  same  side  of  Saturn,  as  seen 
from  the  earth,  it  is  inferred  that  this  satellite  revolves  on  its  axis  once 
during  its  sidereal  period. 

§  507.  The  next  in  order,  proceeding  inwardly,  is  so  obscure  as  to  have 
eluded  the  observations  of  astronomers  until  very  recently.  It  was  dis- 
covered simultaneously  by  Mr.  Bond,  of  Cambridge,  U.  S.,  and  Mr.  Lassell, 
of  Liverpool,  England,  in  1848. 

§  508.  The  next  in  order,  proceeding  in  the  same  direction,  is  by  far 
the  largest  and  most  conspicuous  of  all,  and  probably  not  inferior  to  Mars 
in  size. 

§  509.  The  next  three  in  order  are  very  small,  and  require  pretty  pow- 
erful telescopes  to  see  them,  while  the  two  interior,  which  just  skirt  the 
edge  of  the  ring,  can  only  be  seen  with  telescopes  of  extraordinary  power 
and  perfection,  and  under  the  most  favorable  atmospheric  circumstances." 
When  first  discovered,  they  appeared  to  thread  the  excessively  thin  film 
of  light  reflected  from  the  edge  of  the  ring  then  turned  towards  the  earth, 
and  for  a  short  time  to  advance  off  at  either  end,  speedily  to  return  again. 

§  5 1 0.  Owing  to  the  obliquity  of  their  orbits  to  the  plane  of  Saturn's 
ecliptic,  there  are  no  eclipses,  occultations,  or  transits  of  the  satellites,  or 
shadows  on  the  disk  of  the  primary,  except  at  the  time  when  the  ring  is 
seen  edgewise,  and  their  observation  is  attended  with  too  much  difficulty 
to  be  of  any  practical  use,  like  the  corresponding  phenomena  of  Jupiter's 
satellites,  for  the  determination  of  terrestrial  longitude. 

§  511.  The  names  and  elements  of  Saturn's  satellites  are  given  in  the 
following 

Tabk. 


Names  and 
Order  of 

Satellites. 

Sidereal  Period. 

Mean 
Distance. 

Epoch 
of  Ele- 
ments. 

Mean  Longi- 
tude at  the 
Epoch. 

Eccentri- 
city. 

Perisatur- 

ninii. 

1.  Mimas  .  .  . 

d.    h.    m.    a. 
0  22  37  22.9 

3.3607 

1790.0 

O          '        " 

256  58  48 

2.  Enceladus 

1  08  53  06.7 

4.3125 

1836.0 

67  41  36 

3.  Tethys.  .. 

1  21  18  25.7 

5.3396 

« 

313  43  48 

0.04? 

54°  ? 

4.  Dione.  .  .  . 

2  17  41  08.9 

6.8398 

« 

327  40  48 

0.02? 

42    ? 

5.  Rhea  

4  12  25  10.8 

9.5528 

<( 

353  44  00 

0.02? 

95    I 

6.  Titan  .... 

15  22  41  25.2 

22.1450 

1830.0 

137  21  24 

0.029314256°  38'.  11 

7.  Hyperion. 

22  12    ?      ? 

28.  ± 

8.  lapetus  .  . 

79  07  53  40.4 

64.3590 

1790.0 

269  37  48 

The  longitudes  are  reckoned  in  the  plane  of  the  ring  from  its  descend- 
ing node  on  the  ecliptic.  The  apsides  of  Titan  have  a  direct  motion  of 
30'  23"  per  annum  in  longitude  on  the  ecliptic. 


130 


SPHERICAL    ASTRONOMY. 


§  512.  The  periodic  times  of  the  first  four  satellites  in  order  of  distance 
from  Saturn  are  connected  by  this  law,  viz. :  The  period  of  the  third  is 
double  that  of  the  first,  and  the  period  of  the  fourth  is  double  that  of  the 
second ;  the  coincidence  being  exact  to  within  ^J^  part  of  the  larger 
period. 

SATELLITES  OF  URANUS. 

§  513.  Uranus  is  believed  to  have  six  satellites,  which  revolve  about  the 
primary  from  east  to  west,  in  orbits  nearly,  if  not  quite,  circular,  and  which 
make  with  the  ecliptic  an  angle  of  78°  58'.  They  thus  differ  from  all  the 
other  known  bodies  of  the  solar  system  both  in  the  direction  of  their  mo- 
tion and  inclination  of  their  orbits,  which  latter,  as  well  as  the  places  of 
the  nodes,  have  undergone  no  sensible  change,  during  at  least  one-half  of 
the  planet's  period  around  the  sun. 

The  elements  of  these  satellites,  as  far  as  known,  are  given  in  this 

Table. 


Sat 

Sidereal  Revolution. 

Mean 
Distance. 

Epoch  of  passing  Ascend- 
ing Node. 

Nodes  and  Inclination. 

(Jr.  T. 

1 

4d?h 

Inclination  of  orbits  to 

2 
3 

8  16h  56<n31".3 
10  23? 

17  0 

19  8? 

1787.  Feb.  16,  0^  10m 

the  ecliptic,  78°  58'  ; 
ascending  node  in 
longitude,  165°  30'. 

4 

13  11     07  12.6 

22  8 

1787.  Jan.     7,  Oh  28m 

(Equinox  of  1798.) 

5 
6 

38    2      ? 
107  12      » 

45  5? 
91  0? 

Motion  retrograde, 
and  orbits  nearly 
circular. 

§  514.  The  satellites  of  Uranus  require  very  powerful  and  perfect  tele- 
scopes for  their  observation.  The  second  and  fourth  are  far  the  most  con- 
spicuous, and  their  periods  and  distance  have  been  ascertained  with  toler- 
able certainty.  The  first  and  third  have  also  been  observed  since  their 
original  announcement,  but  of  the  existence  of  the  fifth  and  sixth  we  have 
not  the  same  evidence.  Sir  John  Herschel  is  of  opinion  that  if  future 
observations  should  assign  them  places,  they  would  be  exterior  to  that  of 
the  fourth. 

515.  When  the  earth  is  in  the  plane  of  the  orbits  or  nearly  so,  the  ap- 
parent paths  of  the  satellites  are  straight  lines  or  very  elongated  ellipses, 
in  which  case  these  secondaries  become  invisible  long  before  they  come 
up  to  the  disk  of  the  planet,  in  consequence  of  the  superior  light  of  the 
latter,  so  that  it  is  not  possible  to  observe  their  occultations.  eclipses,  anO 
transits. 


COMETS. 


SATELLITES  OF  NEPTUNE. 


137 


§  516.  If  the  observation  of  the  satellites  of  Uranus  be  difficult,  those 
of  Neptune,  owing  to  the  great  distance  of  this  planet,  must  offer  still 
greater  difficulties.  Of  the  existence  of  one  satellite  there  remains  no 
doubt.  Its  sidereal  period  about  the  planet  is  nearly  5.9  days  ;  its  mean 
distance  is  fourteen  times  Neptune's  semi-diameter ;  and  its  orbit  is  in- 
clined to  the  plane  of  the  ecliptic  under  an  angle  of  about  35°. 


COMETS. 

§  517.  Comets  differ  from  all  the  primary  bodies  with  which  we  have 
thus  far  been  concerned,  in  their  appearance,  the  shape  and  inclination  of 
their  orbits,  and  in  following  no  rule,  as  a  class,  with  regard  to  the  direc- 
tion of  their  motions.  They  are  of  various  sizes,  some  being  visible  to  the 
naked  eye  even  in  daytime,  while  others  require  the  aid  of  telescopes  even 
at  night  to  see  them. 

§  518.  The  larger  consist  for  the  most  part  of  an  ill-defined  mass,  called 
the  head,  from  which,  in  a  direction  opposite  the  sun,  proceeds  a  train,  of 
greater  or  less  extent,  called  the  tail. 

Fig.  94. 


§  519.  The  head  is  much  brighter  towards  its  centre.  Sometimes  this 
increase  of  illumination  terminates  in  a  bright  spot,  called  a  nucleus,  the 
surrounding  haze  which  makes  up  the  rest  of  the  head  being  called  the 
coma. 

§  520.  The  tail  appears  to  consist  of  two  streams  of  luminous  matter 
which,  starting  from  a  point  near  the  head,  and  on  the  side  towards  the 
sun,  pass  suddenly  to  the  opposite  side,  and  grow  broader  and  more  dif- 
fused as  they  increase  in  length ;  they  commonly  unite  at  a  little  distance 
from  the  head,  but  sometimes  continue  distinct  for  the  greater  part  of  their 
course.  This  appendage  has  been  known  to  attain  the  enormous  length 
of  forty-one  millions  of  miles,  and  to  stretch  over  104  degrees  of  the  celes- 
tial sphere. 

§  521.  The  tail  is  not,  however,  an  invariable  appendage  of  comets, 


138  SPHERICAL    ASTRONOMY. 

many  of  the  brightest  having  been  seen  with  little  or  none,  and  others  as 
round  and  well-defined  as  Jupiter. 

§  522.  On  the  other  hand,  there  are  instances  of  comets  with  many 
tails  or  streamers,  spreading  out  like  an  immense  fan,  and  extending  to  the 
distance  of  some  30  degrees  of  the  celestial  vault.  One  is  recorded  as 
having  two  tails,  making  with  each  other  an  angle  of  160°,  the  fainter 
being  turned  towards,  the  other  from  the  sun. 

The  tails  are  often  curved,  bending,  in  general,  towards  that  part  of 
space  which  the  comet  has  left,  as  if  retarded  by  the  opposition  of  some 
resisting  medium. 

§  523.  The  smaller  comets,  such  as  are  only  visible  through  telescopes, 
and  which  are  by  far  the  most  numerous,  present  no  appearance  of  a  tail, 
and  seem  as  round  or  oval  vaporous  masses,  more  luminous  towards  the 
centre,  where,  in  some  instances,  a  small  stellar  point  has  been  seen,  but 
without  any  distinct  nucleus  or  other  signs  of  a  solid  body.  Stars  of  the 
smallest  magnitude,  such  as  would  be  obliterated  by  a  moderate  fog,  are 
seen  through  their  brightest  part. 

§  524.  A  comet  never  exhibits  the  least  signs  of  phases ;  but,  on  the 
contrary  appears  as  a  mass  of  thin  vapor,  either  self-luminous,  or  easily 
penetrated  by  the  luminous  waves  from  the  sun,  which  are  reflected  from 
its  interior  parts  as  from  its  exterior  surface. 

§  525.  The  tail,  where  it  comes  up  and  surrounds  the  head,  is  yet  sep- 
arate from  the  latter  by  an  interval  less  luminous,  as  if  sustained  and  kept 
from  contact  by  a  transparent  stratum  of  atmosphere ;  and  seems  to  be  a 
kind  of  hollow  envelope  of  a  parabolic  form,  inclosing  the  head  near  its 
vertex. 

§  526.  The  number  of  recorded  comets  is  very  great,  amounting  to  sev- 
eral hundred;  and  when  it  is  considered  that  in  the  earlier  stages  of  as- 
tronomy, before  the  invention  of  the  telescope,  only  large  and  conspicuous 
ones  could  be  noticed,  and  that,  since  due  attention  has  been  paid  to  the 
subject,  scarcely  a  year  passes  without  the  observation  of  one  or  two  of 
these  bodies,  and  sometimes  two  or  three  have  appeared  at  once,  it  may 
very  reasonably  be  supposed  that  many  thousands  exist  Multitudes  must 
escape  observation  by  reason  of  their  paths  traversing  only  that  part  of  the 
heavens  which  is  above  the  horizon  in  daytime.  Comets  so  circumstanced 
can  only  become  visible  during  a  total  eclipse  of  the  sun — a  coincidence 
which  is  related  to  have  taken  place  sixty  years  before  Christ,  when  a 
a  large  comet  was  observed  near  the  sun. 

§  527.  The  motion  of  comets  is  characterized  by  the  greatest  irregular- 
ity. Sometimes  they  appear  in  sight  for  a  few  days  only,  at  others  for 


COMETS.  139 

many  months.  Some  move  very  slowly,  others  with  vast  velocity ;  and 
not  unfrequently  the  two  extremes  of  speed  are  exhibited  by  the  same  in- 
dividual in  different  parts  of  its  path.  Some  pursue  a  direct,  others  a  ret- 
rograde, and  others  a  tortuous  and  very  irregular  course ;  nor  are  they 
confined,  like  the  planets,  to  any  particular  region  of  the  heavens,  but 
traverse  indifferently  every  part  alike. 

§  528.  Their  variations  in  apparent  size,  while  visible,  are  equally  re- 
markable ;  sometimes  they  make  their  appearance  as  faint,  slow-moving 
objects,  with  little  or  no  tail ;  by  degrees  they  accelerate  their  speed,  en- 
large and  extend  their  tail,  which  increases  in  length  and  brightness  till 
they  approach  the  sun  near  enough  to  be  lost  in  his  light.  After  a  time 
they  again  emerge  on  the  opposite  side,  receding  from  the  sun.  It  is  now 
for  the  most  part  they  shine  forth  in  all  their  splendor,  and  display  their 
tails  in  greatest  length  and  development.  As  they  continue  to  recede 
from  the  sun  their  motion  diminishes,  their  tails  subside  about  the  head, 
which  grows  continually  feebler  till  lost  in  the  distance,  from  which  by  far 
the  greater  number  have  never  returned ;  thus  indicating  their  paths  to  be 
along  the  parabola  or  hyperbola. 

§  529.  These  seemingly  irregular  and  capricious  movements  are  fully 
explained  by  the  doctrine  of  universal  gravitation,  and  are  no  other  than 
consequences  of  the  laws  of  elliptic,  parabolic,  or  hyperbolic  motions.  But 
the  physical  changes  of  the  head,  the  process  by  which  it  builds  up  the 
enormous  tail,  takes  it  down  again,  and  wraps  it  as  a  mantle  about  itself; 
the  position  of  the  tail  as  regards  the  direction  of  the  sun,  the  multiplicity 
of  tails,  and  other  physical  phenomena  to  be  noticed  presently,  remain 
without  satisfactory  solution. 

§  530.  The  elements  of  a  comet's  orbit  are  readily  computed  from  three 
observed  places,  exactly  as  in  the  case  of  a  planet ;  and  the  comet  usually 
takes  the  name  of  the  computor  who  thus  first  defines  its  track  through 
the  heavens. 

The  elements  of  a  few  now  reckoned  among  the  perm  an  it  members  of 
the  solar  system,  will  be  found  in  the  following  table : 


140 


SPHERICAL   ASTRONOMY. 


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COMETS. 


141 


§  531.  By  far  the  most  interesting  of  these  comets  is  that  of  Halley. 
Its  last  return  took  place  according  to  prediction  in  1835.  While  yet 
remote  from  the  sun  in  its  approach  to  that  luminary,  its  appearance  was 
that  of  an  oval  nebula  without  tail,  and  having  a  minute  point  of  concen- 
trated light  eccentrically  situated  within.  Soon  its  tail  began  to  be  devel- 
oped, and  increased  rapidly  till  it  reached  its  greatest  length,  about  20 
degrees,  when  it  decreased  with  such  haste  as  to  disappear  entirely  before 
perihelion  passage.  When  the  .ail  first  began  to  form,  the  nucleus  be- 
came much  brighter,  and  threw  out  a  jet  or  stream  of  light  towards  the 
sun.  This  ejection  continued,  with  occasional  intermission,  as  long  as  the 
tail  continued  visible.  Both  the  form  and  direction  of  this  luminous 
stream  underwent  singular  and  capricious  alterations,  the  different  phases 
succeeding  one  another  with  such  rapidity  that  no  two  successive  nights 
presented  the  same  appearance.  At  one  time  the  jet  was  single,  at  others 
fan-shaped,  while  at  others  two,  three,  or  more  jets  were  darted  forth  in 
different  directions,  the  principal  one  oscillating  to  and  fro  on  either  side 
of  the  line  drawn  to  the  sun.  These  jets,  though  very  bright  at  their 
point  of  emanation  from  the  nucleus,  faded  away,  and  became  diffused  as 
they  expanded  into  the  coma,  at  the  same  time  curving  backward  as  if 
thrown  against  a  resisting  medium.  After  its  perihelion  passage,  the 
comet  was  not  seen  for  two  months,  and  at  its  reappearance  presented 
itself  under  a  new  aspect.  There  was  no  longer  a  vestige  of  tail ;  it 
seemed  to  the  naked  eye  a  hazy  star  of  the  fourth  magnitude-,  and  through 
a  powerful  telescope  a  small  round  well-defined  disk,  rather  more  than  2' 
in  diameter,  surrounded  by  a  nebulous  coma  of  much  greater  extent. 
Within  the  disk,  and  somewhat  removed  from  its  centre,  appeared  a  mi- 
nute but  bright  nucleus,  from  which  extended,  in  a  direction  opposite  the 
sun,  a  short  vivid  luminous  ray.  As  the  comet  receded  from  the  sun,  the 
coma  disappeared,  as  if  absorbed  into  the  disk,  which  increased  so  rap- 
idly as  in  one  week  to  augment  its  volume  in  the  ratio  of  40  to  1 .  And 
so  it  continued  to  swell  out,  with  undiminished  rate,  until  from  this  cause 
alone  it  ceased  to  be  visible,  the  illumination  becoming  fainter  as  the 
magnitude  increased.  While  this  increase  of  dimensions  proceeded,  the 
form  of  the  disk  passed,  by  gradual  and  successive  additions  to  its  length 
in  the  direction  opposite  to  the  sun,  to  that  of  a  paraboloid,  the  side  towards 
the  sun  preserving  its  planetary  sharpness,  but  the  base  being  so  faint  and 
ill-defined,  as  to  indicate  that  if  the  process  had  been  continued  with  suffi- 
cient light  to  render  it  visible,  a  tail  would  ultimately  have  been  observed. 
The  parabolic  envelope  finally  disappeared,  and  the  comet  took  its  leave 
as  it  came — a  small  round  nebula,  with  a  bright  point  in  or  near  the 


142  SPHERICAL    ASTRONOMY. 

centre.  Figures  5  to  10  inclusive,  of  plate,  taken  iu  ordeiyshow  some  o( 
the  successive  aspects  of  this  comet  at  its  last  appearance. 

§  532.  Many  other  great  comets  are  recorded,  all  affording  peculiarities 
more  or  less  interesting. 

§  533.  On  comparing  the  intervals  between  the  successive  returns  of 
Encke's  comet,  its  periods  are  found  to  be  continually  shortening  ;  that  is, 
its  mean  distance  from  the  sun,  or  semi-major  axis  of  its  orbit,  diminishes 
by  slow  and  regular  degrees,  and  at  the  rate  of  about  Od.ll  during  each 
revolution.  This  is  attributed  to  the  resistance  of  the  ethereal  medium 
which  fills  the  planetary  space,  and  serves  as  the  medium  for  the  transmis- 
sion of  light.  This  resistance  checks  the  velocity,  diminishes  the  centrifu- 
gal force,  and  gives  to  the  sun  more  effect  in  drawing  the  comet  towards 
itself.  It  will  probably  ultimately  fall  into  that  body.  Like  the  comet  of 
Halley,  its  apparent  diameter  is  found  to  diminish  as  it  approaches  to,  and 
to  increase  as  it  recedes  from  the  sun.  It  has  no  tail,  and  presents  to  the 
view  only  a  small  ill-defined  nucleus,  eccentrically  situated  within  a  more 
or  less  elongated  oval  mass  of  vapors,  being  nearest  to  that  vertex  which  is 
towards  the  sun. 

§  534.  Biela's  comet  is  scarcely  visible  to  the  naked  eye ;  its  orbit 
nearly  intersects  that  of  the  earth,  and  had  the  latter,  at  the  time  of  its 
passage  in  1832,  been  a  month  in  advance  of  its  actual  place,  it  would 
have  passed  through  the  comet. 

At  its  last  appearance  it  separated  itself  into  two  parts,  which  contin- 
ued to  journey  along  together,  side  by  side,  through  an  arc  of  70  degrees 
of  their  orbit,  keeping  all  the  while  within  the  same  field  of  view  of  a  tel- 
escope directed  towards  them.  Both  had  nuclei,  both  had  short  tails  par- 
allel to  one  another,  and  perpendicular  to  their  line  of  junction.  At  first 
the  new  comet  was  extremely  small  and  faint  in  comparison  with  the  old  : 
the  difference  both  in  light  and  size  diminished  till  they  became  equal ; 
after  which  the  new  comet  gained  the  superiority  of  light,  presenting,  ac- 
cording to  Lieut.  Maury,  the  appearance  of  a  diamond  spark.  The  old 
comet  soon,  however,  recovered  its  superiority,  and  the  new  one  began  to 
fade,  till  finally  the  comet  was  seen  single  before  it  disappeared.  While 
this  interchange  of  light  was  going  on,  the  new  comet  threw  ou.t  a  faint 
bridge-like  arch  of  light,  which  extended  from  one  to  the  other.  When 
the  original  comet  recovered  its  superior  brightness,  it  in  its  turn  threw 
forth  additional  rays,  so  as  to  present  the  appearance  of  a  comet  with  three 
tails,  forming  with  one  another  angles  of  about  120°.  The  distance  be- 
tween the  comets  at  one  time  was  about  39  times  the  equatorial  radius  of 
the  earth,  or  less  than  two-thirds  the  distance  of  the  moon  from  the  earth. 


Plate  VII. 


TO  FHOZSTT  PAGE  142. 


COMETS  143 

§  535.  The  orbits  of  comets  being  very  eccentric,  and  inclined  under  all 
sorts  of  angles  to  the  ecliptic,  these  bodies  must  pass  near  to  the  planets, 
and  be  more  or  less  affected  by  their  disturbing  action. 

One  passed  Jupiter  at  the  distance  of  £s-  of  the  radius  of  that  planet's 
orbit,  and  the  earth,  three  years  afterwards,  at  seven  times  the  moon's  dis- 
tance. This  comet  was  found  by  Lexell  to  have  passed  its  perihelion  in 
an  elliptical  orbit,  of  which  the  eccentricity  was  0.7858,  and  with  a  pe- 
riodic time  of  about  five  and  a  half  years,  having,  in  all  probability,  been 
drawn  into  this  path  by  the  perturbating  action  of  Jupiter  and  the  earth  at 
its  previous  visits.  Its  next  return  could  not  be  observed  by  reason  of  the 
relative  places  of  its  perihelion  and  of  the  earth,  and  before  another  revo- 
lution could  be  accomplished,  it  passed  within  the  orbit  of  Jupiter's  fourth 
satellite,  and  has  never  been  seen  since.  The  action  of  Jupiter  doubtless 
changed  its  or&t  into  an  extremely  elongated  ellipse,  or  perchance  into  a 
parabola  or  hyperbola ;  and  what  is  most  remarkable,  none  of  Jupiter'e 
satellites  suffered  any  perceptible  derangement — a  sufficient  proof  of  th« 
small  ness  of  the  comet's  mass. 

§  536.  The  great  number  of  comets  which  appear  to  move  in  para- 
bolic orbits,  or  elliptical  orbits  so  elongated  as  not  to  be  distinguished  from 
them,  has  given  rise  to  an  impression  that  these  bodies  are  extraneous  to 
our  system,  and  that  our  elliptic  comets  owe  their  permanent  denizenship 
within  the  sphere  of  the  sun's  dominant  attraction  to  the  retarding  action 
of  one  or  other  of  the  planets  near  which  they  may  have  passed,  and  by 
which  their  velocity  was  reduced  to  compatibility  with  elliptic  motion. 
A  similar  disturbing  cause,  acting  to  increase  the  velocity,  would  give  rise 
to  a  parabolic  or  hyperbolic  orbit,  so  that  it  is  not  impossible  for  a  comet 
to  be  drawn  into  our  system,  retained  during  many  revolutions  about  the 
sun,  and  finally  expelled  from  it,  never  more  to  return,  as  was  probably 
the  case  with  that  of  Lexell. 

§  537.  The  fact  that  all  the  planets  and  nearly  all  the  satellites  move 
in  one  direction  about  the  sun,  while  retrograde  comets  are  very  common, 
would  go  far  to  assign  them  an  extraneous  origin.  From  a  consideration 
of  all  the  cometary  orbits  known  in  the  early  part  of  the  present  century, 
Laplace  found  that  the  average  situation  of  their  planes  was  so  nearly  per 
pendicular  to  the  ecliptic  as  to  afford  no  presumption  of  any  cause  biasing 
their  inclinations.  And  yet  as  the  planes  of  the  elliptical  orbits  approach 
that  of  the  ecliptic,  the  number  of  direct  comets  increases ;  and  a  plane  of 
motion  coincident  with  that  of  the  earth,  and  periodicity  of  return,  are  de- 
cidedly favorable  to  direct  motion. 


144  SPHERICAL   ASTRONOMY. 

STARS. 

§  538.  Besides  the  bodies  composing  the  solar  system,  there  are  a 
countless  multitude  of  others  which,  because  they  retain  their  relative  places 
sensibly  unchanged  are  called,  though  improperly,  fixed  stars.  Like  our 
sun  they  are  poised  in  space,  are  self-luminous,  and  in  all  probability  are 
centres  of  planetary  systems. 

§  539.  Among  these  stars,  which  at  first  view  seem  scattered  over  the 
celestial  vault  at  random,  appears,  every  evening,  a  bright  band,  called  the 
milky  loay,  that  stretches  from  horizon  to  horizon  and  forms  a  zone  com- 
pletely encircling  the  heavens.  It  divides  in  one  part  of  its  course  into 
two  branches,  which  unite  again  after  remaining  separate  for  150°  of-  their 
course. 

§  540.  The  most  refined  observations  have  been  able  to  assign  to  none 
of  the  stars  a  sensible  geocentric,  and  to  but  very  few  only  an  exceeding 
small  and  uncertain  annual  parallax  •  while  the  most  powerful  magnifiers 
have  thus  far  failed  to  reveal  an  appreciable  disk. 

But  little  can,  therefore,  be  known  of  their  distances,    nothing  at  all  of 
their  real  dimensions,  and  the  only  means  by  which  one  may  be  distin 
guished  from  another  are  in  the  character  and  intensity  of  their  illumina 
tion. 

§  541.  It  is  usual  to  arrange  the  stars  into  classes  called  magnitudes, 
and  this  without  reference  to  their  location  in  the  heavens.  The  brightest 
are  said  to  be  of  the  first  magnitude,  those  which  fall  so  far  short  of  the 
first  degree  of  brightness  as  to  make  a  strongly  marked  distinction,  are 
classed  in  the  second,  and  so  on  down  to  the  sixth  or  seventh,  which  com- 
prise the  smallest  stars  visible  to  the  naked  eye  in  the  clearest  and  darkest 
night. 

§  542.  Beyond  this,  however,  telescopes  continue  the  range  of  visibility 
down  to  the  16th;  nor  does  there  seem  any  reason  to  assign  a  limit  to  the 
progression,  for  every  increase  in  the  dimensions  and  po*ver  of  telescopes 
has  brought  into  view  multitudes  innumerable  of  objects  invisible  before ; 
and,  for  any  thing  experience  has  taught  us,  the  number  of  stars  may,  to 
our  powers  of  enumeration,  be  regarded  as  absolutely  without  limit. 

§  543.  The  mode  of  classification  into  orders  is  entirely  arbitrary.  Of 
a  multitude  of  bright  objects,  differing  in  all  probability  intrinsically  both 
in  size  and  splendor  and  arranged  at  unequal  distances,  one  must  appear 
the  brightest,  another  next  below  it,  and  so  on.  An  order  of  succession 
must  exist,  and  when  it  is  gradual  in  degree  and  indefinite  in  extent,  tc 
draw  a  line  of  demarkation  is  matter  of  pure  convention. 


STARS. 


145 


§  544.  Sir  John  Herschel  proposes  to  make  the  scale  of  decreasing 
brightness  of  the  stars  which  head  the  several  ordeis  of  magnitudes,  to 
vary  inversely  as  the  squares  of  the  natural  numbers,  or  as  1,  J-,  £,  TL,  ^-, 
<fec. ;  that  is,  the  brightest  star  of  the  first  magnitude  shall  be  four  times 
that  of  the  brightest  of  the  second,  nine  times  that  of  the  brightest  of  the 
third,  and  so  on:  stars  of  intermediate  brightness  to  be  expressed  deci- 
mally. Thus  a  star  half  way  in  brightness  between  the  brightest  of  the 

third  and  of  the  fourth  magnitudes  would  be  expressed  by -^ 

On  the  hypothesis  that  all  the  stars  possess  the  same  intrinsic  brightness, 
coupled  with  the  fact  that  the  distance  of  the  same  luminous  object  varies 
inversely  as  the  square  root  of  its  Apparent  brightness,  the  mere  mention  of 
the  magnitudes  of  the  stars  would  suggest,  according  to  this  classification, 
their  relative  distribution  through  space. 

§  545.    To  accomplish   this  Fi&  «*. 

photometrical  classification,  he 
proposes  to  receive  the  light  from 
the  planet  Jupiter,  at  A,  on  the 
first  face  of  a  triangular  prism, 
«*o  as  to  fall  on  the  second  face 
/it  C  under  an  angle  of  total 
reflection;  this  light,  on  its 
emergence  from  the  third  face, 
being  received  upon  a  convex 
lens  Z>,  would  form  an  image 
of  Jupiter's  disk  at  F.  An  eye 
placed  at  E,  within  the  field  of 
the  diverging  waves,  would  re- 
ceive the  light  from  this  image 
and  that  from  a  star  proceeding 
along  the  line  BE.  The  ap- 
parent brightness  of  Jupiter's 
image  would  vary  inversely  as 
the  square  of  FE,  because  this 

planet  has  no  sensible  phases,  and  under  the  same  atmospheric  circum- 
stances is  of  a  constant  brightness,  while  that  of  the  star  would  be  constant 
for  all  positions  of  the  eye,  and  by  altering  the  place  of  the  latter  the  star 
and  the  image  may  be  made  to  appear  equally  bright.  The  value  of  EF 
being  ascertained  for  different  stars,  their  relative  brightness  becomes 
known. 

10 


SPHERICAL    ASTRONOMY. 

§  546.  Astronomers  have  generally  agreed  to  restrict  the  first  magni- 
tude to  about  23  or  24  stars,  the  second  to  50  or  60,  the  third  to  about 
200,  and  so  on,  their  numbers  increasing  rapidly  as  we  proceed  in  the  order 
of  decreasing  brightness,  the  number  of  stars  registered  to  include  the  sev 
enth  magnitude  being  from  12  to  15  thousand. 

§  547.  Stars  of  the  first  three  or  four  magnitudes  are  distributed  pretty 
uniformly  over  the  celestial  sphere,  the  number  being  somewhat  greater, 
however,  especially  in  the  southern  hemisphere,  along  a  zone  following  the 
course  of  a  great  circle  through  the  stars  called  s  Orionis  and  a  Ousis. 
But  when  the  whole  number  visible  to  the  naked  eye  are  considered,  they 
increase  greatly  towards  the  borders  of  the  milky  way.  And  if  the  tele- 
scopic stars  be  included,  they  will  be  foiind  crowded  beyond  imagination 
along  the  entire  extent  of  tliat  remarkable  belt  and  its  branches.  Indeed, 
its  whole  light  is  composed  of  stars  of  every  magnitude  from  such  as  are 
visible  to  the  naked  eye  to  the  smallest  point  perceptible  through  the  be«t 


§  548.  The  general  course  of  the  milky  way,  neglecting  occasional  de- 
viations and  following  the  greatest  brightness,  is  that  of  a  great  circle  in- 
clined to  the  equinoctial  under  an  angle  of  63°,  and  cutting  that  circle  in 
right  ascension  Oh  47m  and  12h  47m,  so  that  its  northern  and  southern  poles 
are  respectively  in  right  ascension  18h  47m  and  6h  47m. 

§  549.  This  great  circle  of  the  celestial  sphere  with  which  the  general 
course  of  the  milky  way  most  nearly  coincides,  is  called  the  gallactic  circle. 
To  count  the  number  of  stars  of  all  magnitudes  visible  in  a  single  field  of 
a  telescope,  and  to  alter  the  field  so  as  to  take  in  successively  the  entire 
celestial  sphere,  is  to  gauge  the  heavens. 

§  550.  A  comparison  of  many  different  gauges  has  given  the  average 
number  of  stars  in  a  single  field  of  15'  diameter,  within  zones  encircling 
the  poles  of  the  gallactic  circle,  found  in  the  following 

Table. 

Zones  of  North  Gallactic  Average  Number  ol  Stars 

Polar  distance.  in  field  of  15'. 

0°  to  15°  .  .  .  4.32 

15  to  30  .  .  .  .  5.42 

30  to  45  .  .  .  .  8.21 

45  to  60  .  .  .  .  13.61 

60  to  75  .  .  .  24.09 

75  to  90  53.43 


STARS.  14.7 

Zones  of  South  Gallactic  Average  Number  of  St*r» 

Polar  distance.  it  field  of  15'. 

0°   to  15°  '.  .  .  .  6.05 

15     to  30  .  .  .  .  6.62 

30     to  45  .  .  .  .  9.08 

45     to  60  .  .  .  .  13.49 

60     to  75  .  .  .  .  26.29 

75     to  90  .  .  .  .  59.06 

§  551.  This  shows  that  the  stars  of  our  firmament,  instead  of  being 
scattered  in  all  directions  indifferently  through  space,  form  a  stratum  of 
which  the  thickness  is  small  in  comparison  with  its  length  and  breadth, 
and  that  our  sun  occupies  a  place 
somewhere    about  the  middle   of 
the  thickness,  and  near  the  point 
where  it  subdivides  into  two  prin- 
cipal   laminae,   inclined    under   a 
small  angle  to  one  another.     For 
to  an  eye  so  situated,  the  apparent 
density  of  stars,  supposing  them 
pretty  equally  scattered   through 

the  space  they  occupy,  would  be  least  in  the  direction  A  S,  perpendicular 
;o  the  laminae,  and  greatest  in  that  of  its  breadth  S  B,  S  C,  or  SD  ;  in- 
creasing rapidly  in  passing  from  one  direction  to  the  other. 

§  552.  For  convenience  of  reference  and  of  mapping,  the  stai's  are  sep- 
arated into  groups  by  conceiving  inclosing  lines  drawn  upon  the  celestial 
sphere  after  the  manner  of  geographical  boundaries  on  the  earth.  The 
groups  of  stars  within  such  boundaries  are  called  constellations  The 
brightest  star  in  each  constellation  is  designated  by  the  first  letter  of  the 
Greek  alphabet,  the  next  brightest  by  the  second,  and  so  on  till  this  alpha- 
bet is  exhausted,  when  recourse  is  had  to  the  Roman  alphabet,  and  then  to 
numerals.  A  star  will  be  known  from  the  name  of  the  constellation  and 
the  letter  or  numeral  :  thus,  a  Centauri,  61  Cygni.  Many  of  the  bright- 
est stars  have  also  proper  names,  as  Sirius,  Arcturus,  Polaris,  <fcc. 

§  563.  If,  in  Eq.  (28),  p  denote  the  radius  of  the  earth's  orbit,  «•  becomes 
the  annual  parallax,  d  the  star's  distance,  and  w  as  before  the  number  of 
seconds  in  radius  unity.  That  equation  gives 

-  ........  (ieo) 


§  554.  A  line  connecting  the  earth  and  a  star  would  in  the  course  of  a 
year  describe  the  entire  surface  of  a  cone  of  which  the  vertex  would  be  the 


SPHERICAL    ASTRONOMY. 

star,  and  the  base  the  orbit  of  the  earth.  The  intersection  of  the  nappe  of 
this  cone  beyond  the  star  with  the  celestial  sphere  would  be  an  ellipse,  and 
the  apparent  orbit  of  the  star,  arising  from  heliocentric  parallax.  The 
greater  axis  of  this  ellipse  would  be  double  the  annual  parallax. 

§  555.  The  stars  floating,  as  it  were,  in  space,  and  being  subjected  to 
the  laws  of  universal  gravitation,  must  each  have  a  proper  motion.  In  con- 
sequence of  their  vast  distances  from  one  another  this  motion  ma}7  be  com- 
paratively slow,  and  their  excessive  distance  from  us  almost  conceals  it,  re- 
quiring years  to  describe  spaces  sufficiently  great  to  subtend  sensible  angles 
at  the  earth.  By  comparing  the  relative  places  of  stars  at  remote  periods 
this  proper  motion  has  been  detected  and  measured  in  a  great  many  in- 
stances. 

§  556.  Stars  having  the  greatest  proper  motion  are  inferred  to  be  near- 
est to  us,  and  this  has  determined  the  selection  of  certain  stars  in  preference 
to  others  in  the  efforts  which  have  been  made  to  ascertain  their  paral- 
hxes. 

§  557.  Two  methods  have  been  pursued.  First,  to  find  by  careful  me- 
ridional observations  of  right  ascensions  and  declinations,  cleared  from  re- 
fraction, nutation,  aberration,  and  proper  motion,  the  places  of  the  star 
throughout  the  year,  and  thence  the  distance  between  those  places  most 
remote  from  one  another.  This  is  double  the  annual  parallax. 

Second,  after  selecting  two  stars  very  near  10  one  another,  and  of  which 
one  has  an  obvious  proper  motion  and  the  other  not,  to  measure  with  the 
heliometer  or  micrometer  their  apparent  distances  apart,  and  to  note  the 
corresponding  positions  of  the  line  joining  them  throughout  the  year;  then 
to  construct  therefrom,  after  correcting  for  proper  motion,  the  annual  path 
of  the  moving  star.  Its  longer  axis  will  be  double  the  annual  parallax. 
This  second  is  greatly  the  preferable  method.  The  stars  being  separated 
by  a  few  seconds  only,  they  will  be  equally  affected  by  refraction,  nutation, 
and  aberration,  none  of  these  depending  upon  actual  distance.  The  method 
supposes  the  apparently  immovable  star  to  be  immensely  distant  beyond 
the  movable  one. 

By  the  first  method  Professor  Henderson  found  the  parallax  of  a  Cen- 
tmiri  to  be  0  ".913  ;  and  by  the  second  M.  Bessel  that  of  61  Cygni  to  be 
0".348. 

§  558.  Assuming  the  parallax  of  a  Centauri  =  1",  to  avoid  multiplicity 
of  figures,  substituting  it  for  <K  in  Eq.  (160),  and  writing  the  numerical 
value  of  w,  we  have 

d       u 

-  =  -  =  203265  ....         (161) 


STARS.  149 

and  in  this  proportion  at  least  must  the  distance  of  the  fixed  stars  exceed 
the  distance  of  the  sun  from  the  earth. 

Substituting  for  p  its  value,  say  in  round  numbers  95,000,000  of  miles, 
and  we  have 

d  -  206265  X  95000000  =  195951 75000000"1, 

or  about  twenty  billions  of  miles. 

§  559.  Denoting  the  velocity  of  light  by  v,  the  time  required  for  it  to 
traverse  the  distance  which  separates  the  star  from  the  earth  by  t,  we  have 
first,  §  505, 

v  =  192000m, 
arid 

t  =  -  —  3y.23  ; 

that  is  to  say,  it  would  require  light  three  years  and  a  quarter  to  come 
from  the  nearest  fixed  star  to  the  earth.  And  as  this  is  the  inferior  limit 
which  it  is  already  ascertained  that  even  the  brightest  and  therefore,  in  the 
absence  of  all  other  indications,  the  nearest  stars  exceed,  what  is  to  be  al- 
lowed for  the  distances  of  those  innumerable  stars  of  the  smaller  magni- 
tudes which  the  most'  powerful  telescopes  disclose  in  the  remote  regions  of 
the  milky  way  ? 

§  560.  The  space  penetrating  power  of  a  telescope,  or  the  comparative 
distance  to  which  a  star  would  require  to  be  removed  in  order  that  it  may 
appear  of  the  same  brightness  through  the  telescope  as  it  did  before  to  the 
naked  eye,  may  be  calculated  from  the  aperture  of  the  telescope  as  com- 
pared with  that  of  the  pupil  of  the  eye,  and  from  its  power  of  reflecting  or 
of  transmitting  incident  light.  The  space  penetrating  power  of  the  tele- 
scope employed  on  the  gauge  stars  referred  to  in  §  550  was  75.  A  star 
of  the  6th  magnitude  removed  to  75  times  its  distance  would  therefore  still 
be  visible,  as  a  star,  through  that  instrument,  and  admitting  such  a  star  to 
have  100th  part  the  light  of  a  standard  star  of  the  1st  magnitude,  it  will 
follow,  from  the  law  of  illumination  and  distance,  that  such  standard  star 
if  removed  75  x  10  =  750  times  its  distance  would  excite  in  the  eye, 
when  viewed  through  the  telescope,  the  same  impression  as  a  star  of  the 
6th  magnitude  does  in  the  naked  eye.  Among  the  infinite  number  of 
stars  in  the  remoter  regions  of  the  milky  way  it  is  but  reasonable  to  con- 
clude that  there  are  many  individuals  intrinsically  as  bright  as  those  which 
immediately  surround  us.  The  light  of  such  stars  must,  therefore,  have 
occupied  750  X  3.25  =  2437.5  years  in  travelling  over  the  distance 
which  separates  them  from  our  own  system.  And  it  follows  that  when 
we  observe  the  places  and  note  the  appearances  of  such  stars,  we  are  only 


150  SPHERICAL   ASTRONOMY. 

reading  their  history  more  than  two  thousand  years  before.  Nor  is  this 
conclusion,  startling  as  it  may  appear,  to  be  avoided  without  attributing 
HU  inferiority  of  intrinsic  illumination  to  all  the  stars  of  the  milky  way — 
an  alternative  much  less  in  harmony,  as  we  shall  see  presently,  with  astro 
nomical  facts  connected  with  other  sidereal  systems,  revealed  by  the  tele- 
scope, than  are  the  views  just  taken. 

§  561.  Of  some  of  the  stars  whose  parallaxes  have  been  determined, 
the  values  of  the  parallaxes,  and  the  names  of  the  discoverers,  are  given  in 
this 

Table. 

a  Centauri       .         .         .         .  0.913;  Henderson. 

61  Cygni  ....  0.348;  Bessel. 

n  Lyra  ....  0.261 ;  Struve. 

Sirius  ....  0.230;  Henderson. 

1831   Groombridge         .         .         .  0.226;  Peters, 

i  Ursge  Majoris         .         .         .  0.133;         " 

Arcturus       .         .  .  0.127;         " 

Polaris          ....  0.067;         " 

Capella         .         .         .  0.046; 

§  562.* As  remarked  in  the  beginning  of  this  chapter,  the  very  best 
t.elescopes  afford  only  negative  information  respecting  the  apparent  diam- 
eters of  the  stars.  The  round  and  well-defined  planetary  disks  which  good 
telescopes  exhibit  are  mere  optical  illusions,  these  disks  diminishing  more 
and  more  in  proportion  as  the  aperture  and  power  of  the  instrument  are 
increased.  And  the  strongest  evidence  of  a  total  absence  of  perceptible 
dimensions  is  the  fact,  that  in  occultations  of  the  stars  by  the  moon,  the 
extinctions  are  absolutely  instantaneous. 

If  our  sun  were  removed  to  the  distance  of  a  Centauri,  its  apparent  di- 
ameter of  32'  3"  would  be  reduced  to  only  0".0093,  a  quantity  which  no 
improvement,  of  OUT  present  instruments  can  ever  show  with  an  apprecia- 
ble disk. 

§  563.  The  star  a,  Centauri  has  been  directly  compared  with  the  moon 
by  the  method  of  §  545.  By  eleven  such  comparisons,  after  making  due 
allowances  for  known  sources  of  error,  it  was  found  that  the  light  of  the 
full  moon  exceeded  that  of  the  star  in  the  proportion  of  27408  to  1. 
Wollaston  found  the  proportion  of  the  sun's  light  to  that  of  the  moon  tc 
be  as  801072  to  1.  Combining  these  results,  the  light  we  receive  from 
the  sun  is  to  that  from  a  Centauri  as  21,955,000,000.  or  about  twenty- 
two  thousand  millions  to  one.  Hence,  the  illumination  being  inversely  as 


STARS.  151 

fche  sq  lare  of  the  distance,  the  intrinsic  splendor  of  tHs  star  is  to  that  of 
the  sun  as  2.3247  to  1.  The  light  of  Sirius  is  four  times  that  of  a  Cen~ 
tauri,  and  its  parallax  only  0".230,  which  give  to  Sirius  a  splendor  equal 
to  140.2  times  that  of  the  sun. 

§  564.  Periodical  Stars. — Many  of  the  stars,  which  in  other  respects 
are  no  way  distinguished  from  the  rest,  undergo  periodical  increase  and 
diminution  of  brightness,  involving  in  one  or  two  instances  complete  ex- 
tinction and  renovation.  These  are  called  periodical  stars. 

§  565.  The  most  remarkable  star  in  this  respect  is  o  Ceti,  sometimes 
called  Mira.  It  appears  at  variable  intervals,  of  which  the  mean  is  33  ld 
I5h  7m.  It  retains  its  greatest  brightness  for  a  fortnight,  being  on  some 
occasions  equal  to  a  large  star  of  the  second  magnitude ;  decreases  for 
about  three  months,  becoming  completely  invisible  to  the  naked  eye  for 
about  five  months,  and  increases  for  the  remainder  of  the  period.  Such  is 
the  general  course  of  its  phases.  It  does  not  always  return  to  the  same 
degree  of  brightness,  nor  increase  nor  decrease  by  the  same  gradations, 
neither  are  the  successive  intervals  of  maxima  equal.  The  mean  interval 
is  subject  to  a  cyclical  fluctuation  embracing  eighty-eight  such  intervals, 
and  having  the  effect  to  shorten  and  lengthen  the  same  about  25  days  one 
way  and  th:-  other. 

§  566.  Another  very  remarkable  periodical  star  is  that  called  j8  Persei, 
and  also  frequently  called  Algol.  It  is  usually  visible  as  a  star  of  the 
second  magnitude,  and  as  such  continues  for  2d  13h.5,  when  it  suddenly 
begins  to  diminish  in  splendor,  and  in  about  3h.5  is  reduced  to  the  fourth 
magnitude,  at  which  it  continues  for  about  I5m.  It  then  begins  to  in- 
crease, and  in  3h.5  is  restored  to  its  usual  brightness,  going  through  all  its 
changes  in  2d  20h  48m  58g.5.  Recent  observations  indicate  that  this  period 
is  on  the  decrease,  and  not  uniformly,  but  with  an  accelerated  rapidity, 
indicating  that  it  too  has  its  cyclical  period,  and  that  instead  of  continuing 
to  decrease,  it  will  after  a  while  be  found  to  increase. 

§  567.  The  star  8  Cepheus  is  also  a  periodical  star.  Its  period  from 
minimum  to  minimum  is  5d  8h  47m  39".5.  The  extent  of  its  variations  is 
from  the  fifth  to  between  the  third  and  fourth  magnitudes.  Its  increase  is 
more  rapid  than  its  diminution- -the  former  occupying  ld  14h,  and  the 
latter  3d  19". 

§  568.  The  periodical  star  $  Lyra  has  a  period  of  12d  21h  53m  10s, 
within  which  a  double  maxima  and  minima  take  place,  the  maxima  being 
about  equal,  but  the  minima  not  The  maxima  are  about  3.4,  and  the 
minima  4.3  and  4.5.  Here  again  the  period  is  subject  to  change,  which 
.;«  itself  periodical. 


153  SPHERICAL    ASTRONOMY. 

§  569.  Numerous  other  periodical  stars  are  recorded.  These  remark- 
able variations  of  brightness,  and  the  laws  of  their  periodicity,  have  sug- 
gested the  revolution  of  some  opaque  body  or  bodies  around  the  stars  thus 
distinguished,  which,  becoming  interposed  at  inferior  conjunction,  would 
intercept  a  greater  or  less  portion  of  the  light  on  its  way  to  the  earth.  Or 
the  stars  may  possess  very  different  degrees  of  intrinsic  illumination  on 
different  portions  of  their  surfaces,  which,  being  subject  to  periodical 
changes  and  presented  to  the  earth  by  an  axial  rotation  of  the  stars, 
would  produce  the  phenomena  in  question. 

§  570.  Temporary  Stars. — The  irregularities  above  referred  to  may 
afford  an  explanation  of  other  stellar  phenomena,  which  have  hitherto 
been  regarded  as  altogether  casual.  Stars  have  appeared  from  time  to 
time  in  different  parts  of  the  heavens  blazing  forth  with  extraordinary 
splendor,  and  after  remaining  a  while,  apparently  immovable,  have  faded 
away  and  disappeared.  These  are  called  temporary  stars.  One  of  these 
stars  is  said  to  have  appeared  about  the  year  125  B.  c.,  and  with  such 
brightness  as  to  be  visible  in  the  daytime.  Another  appeared  in  A.  D.  389, 
near  a  Aquilse,  remaining  for  three  weeks  as  bright  as  Venus,  and  disap- 
pearing entirely.  Also  in  945,  1264,  and  1572,  brilliant  stars  appeared 
between  Cepheus  and  Cassiopeia,  which  are  supposed  to  be  one  and  the 
same  periodical  star,  with  a  period  of  312,  or  perhaps  156  years.  The 
appearance  in  1572  was  very  sudden.  The  star  was  then  as  bright  as 
Sirius ;  it  continued  to  increase  till  it  surpassed  Jupiter,  and  was  visible  at 
mid-day.  It  began  to  diminish  in  December  of  the  same  year,  and  in 
March,  1574,  it  had  entirely  disappeared.  So,  also,  on  the  10th  of  Octo- 
ber, 1604,  a  star  not  less  brilliant  burst  forth  in  the  constellation  Serpeit 
tarius,  which  continued  visible  till  October,  1605. 

§  571.  Similar  phenomena,  though  of  less  splendor,  have  taken  place 
more  recently.  A  star  of  the  fifth  magnitude,  or  5.4,  very  conspicuous  to 
the  naked  eye,  suddenly  appeared  in  the  constellation  Ophiuchus.  From 
the  time  it  was  first  seen  it  continued  to  diminish,  without  alteration  of 
place,  and  before  the  advance  of  the  season  put  an  end  to  the  observations 
upon  it,  had  become  almost  extinct.  Its  color  was  ruddy,  which  was 
thought  to  have  undergone  many  remarkable  changes. 

§  572.  The  alternations  of  brightness  of  >]  Argus  are  very  remarkable. 
In  1677  it  appeared  as  a  star  of  the  fourth,  in  1751  of  the  second,  in  1811 
and  1815  of  the  fourth,  in  1822  and  1826  of  the  second,  in  1827  of  the 
first,  and  in  1837  of  the  second  magnitude.  All  at  once,  in  1838,  it  sud* 
denly  increased  in  lustre  so  as  to  surpass  all  the  stars  of  the  first  magnitude 
except  Sinus,  Canopus,  and  a  Centauri.  Then  it  again  diminished,  but  not 


STARS. 


153 


below  the  first  magnitude,  till  April,  1843,  when  it  had  increased  so  as  to 
surpass  Canopus,  and  nearly  equal  Sirius. 

§  573.  On  careful  re-examination  of  the  heavens,  and  comparison  of 
catalogues,  many  stars  are  missing. 

§  574.  Double  Stars. — Many  of  the  stars  when  examined  through  the 
telescope  appear  double,  that  is,  to  consist  of  two  individuals  close  to- 
gether. They  are  divided  into  classes  according  to  the  proximity  of  their 
component  individuals.  The  first  class  comprises  those  only  of  which  the 
distance  does  not  exceed  1"  ;  the  second  those  in  which  it  exceeds  1",  but 
falls  short  of  2"  ;  the  third  those  in  which  it  ranges  from  2"  to  4"  ;  the 
fourth  from  4"  to  8"  ;  the  fifth  from  8"  to  12" ;  the  sixth  from  12"  to 
16" ;  the  seventh  from  16"  to  24"  ;  and  the  eighth  from  24"  to  32". 

Each  of  these  classes  is  subdivided  into  two  others,  called  respectively 
conspicuous  and  residuary  double  stars.  The  first  comprehends  those  in 
which  both  individuals  exceed  the  8.25  magnitude,  and  are  therefore  sep- 
arately bright  enough  to  be  seen  with  telescopes  of  very  moderate  capa- 
city ;  the  second  embraces  those  which  are  below  this  limit  of  visibility. 
Specimens  of  each  class  will  be  found  in  the  following 


y  Coronae  Bor. 
y  Centauri. 
y  Lupi. 
c  Arietis. 
t  Herculis. 


Table. 

CLASS  L—  0"  TO  1". 

n  Coronae. 
17  Herculis. 
A  Cassiopeia?. 
A  Ophiuchi. 
v  Lupi. 

7  Ophiuchi. 
p  Draconis. 
<p  Ursae  Majoris. 
£  Aquilae. 
u  Leonis 

Atlas  Pleiadum 
4  Aquarii. 
42  Comae. 
52  Arietis. 
66  Piscium. 


CLASS  II.— 1"  TO  2' 


Y 

Circini. 

5 

Bootis. 

| 

Ursa  Majoris. 

2 

Camelopardi. 

t 

Cygni. 

i 

Cassiopeiae. 

K 

Aquilae. 

82 

Orionis. 

c 

Chamaeleontia 

«a  Cancri. 

a 

Coronae  Bor. 

52 

Orionis. 

CLASS 

III.—  2" 

TO  4". 

• 

Piscium. 

V 

Virginis. 

5 

Aquarii. 

^ 

Draconis. 

0 

Hydras. 

& 

Serpentis. 

5 

Orionis. 

t 

Canis. 

Y 

Ceti. 

t 

Bootis. 

i 

Leonis. 

P 

Heroulis. 

Y 

Leonis. 

t 

Draconis. 

t 

Trianguli. 

ff 

Cassiopeia. 

Y 

Coronse  Aus. 

t 

Hydrae. 

K 

Leporis. 

44 

Bootis. 

CLASS 

IV.—  4"  TO  8". 

a 

Crusis. 

e 

Phoenicia. 

£ 

Cephei. 

ft 

Eridani. 

a 

Herculis. 

K 

Cephei. 

7T 

Bootis. 

70 

Ophiuchi. 

a 

Geminorum. 

A 

Orionis. 

P 

Capricorni. 

12 

Eridani. 

6 

Geminoruin. 

/* 

Cygni. 

V 

Argus. 

82 

Eridani. 

T 

Coronaa  Bor. 

t 

Bootiu. 

w 

Auriga. 

95 

Herculia. 

J54 


SPHERICAL   ASTRONOMY. 


ft  Orionis. 
y  Arietis. 
y  Delphini. 


a  Centauri. 
ft  Cephei. 
0  Scorpii. 


a  Canum  Ven. 
c  Normae. 
5  Piscium. 


i  Herculis. 
n  Lyras. 
<  Cancri. 


CLASS  V.— 8-'  TO  12". 

p  Antilae. 
i)  Cassiopeiae. 
0  Eridani. 

CLASS  VI— 12"  TO  16" 

v  Volantis. 

17  Lupi. 

p  Ursae  Majoris. 

£LASS  VII— 16"  TO  24". 
0  Serpentis. 
K  Coronae  Aus. 
X  Tauri. 

CLASS  VIIL— 24"  TO  32" 

K  Herculis. 
K  Cephei. 
$  Draconis. 


1  Orionis. 
/  Eridani. 

2  Canum  Ven. 


K  Bootis. 
8  Monocerotis. 
61  Cygni. 


24  Comae. 
41  Draconis. 
61  Ophiuchi. 


X  Caucri. 
23  Orionis. 


§  575.  Triple,  Quadruple,  and  Multiple  Stars. — Stars  which  answei 
to  these  designations  also  occur,  and  of  them  the  most  remarkable  are, 


a  Andromedae. 
e  Lyra. 
$  Cancri. 


Q  Orionis. 
//  Lupi. 
Bootis. 


£  Scorpii. 

11  Monocerotis. 

12  Lyncis. 


Of  these,  a,  Andromedce,  fx  Bootis,  and  fjt  Lupi,  appear  through  telescopes 
of  considerable  optical  power  only  as  ordinary  double  stars ;  and  it  is  only 
when  excellent  instruments  are  used  that  their  companions  are  subdivided 
and  found  to  be  extremely  close  double  stars,  e  Lyra  offers  the  remarka- 
ble example  of  a  double-double  star.  In.  telescopes  of  low  power  it  ap- 
pears as. a  coarse  double  star,  but  on  increasing  the  power,  each  individual 
is  perceived  to  be  double,  the  one  pair  being  about  2".5,  the  other  about 
3"  apart.  Each  of  the  stars  £  Cancri,  %  Scorpii,  11  Monocerotis,  and 
12  Lyncis,  consists  of  a  principal  star  closely  double  and  a  smaller  and 
more  distant  attendant;  while  &  Orionis,  (Fig.  11,  of  plate,)  presents  four 
brilliant  principal  stars  of  the  4th,  6th,  7th,  and  8th  magnitudes,  forming 
a  trapezium,  of  which  the  longest  diameter  is  24".4,  and  accompanied  by 
two  excessively  minute  and  very  close  companions,  to  perceive  both  of 
which  is  one  of  the  severest  tests  that  can  be  applied  to  a  telescope. 

§  576.  Of  the  delicate  subclass  of  double  stars,  or  those  consisting  of 
very  large  and  conspicuous  double  stars,  accompanied  by  very  minute 
companions,  the  following  are  specimens,  viz. : 


Plate  VET. 


TO  YB-otrr  PAOE 


STARS.  155 


42  Cancri. 

a  Polaris. 

it  Circini. 

0  Virginia. 

a3  Capricorni. 

0  Aquarii. 

K  Geminorum. 

%  Eridani. 

a    Indi. 

y  Hydree. 

p  Persei. 

16  Aurigae. 

a    Lyra. 

i  Ursa  Major. 

7  Bootis. 

91  Ceti. 

§  577.  Binary  Stars. — Many  of  the  double  stars  are  physically  con- 
lected  in  such  proximity  to  one  another  as  to  revolve  about  their  common 
jentre  of  gravity  in  regular  orbits.  These  are  called  Unary  stars.  They 
liffer  from  what  are  called  ordinarily  "  double  stars"  in  being  so  near  to 
one  another  as  to  be  kept  asunder  only  by  a  rotary  motion  about  a  com- 
mon centre  ;  whereas  the  individuals  of  a  double  star  are  separated  by  a 
vast  distance,  and  appear  double  only  in  consequence  of  one  being  almost 
directly  behind  the  other  as  seen  from  the  earth. 

§  578.  The  position  micrometer  gives  from  time  to  time  the  apparent 
distance  between  the  places  into  which  the  stars  of  a  binary  system  are 
projected  upon  the  celestial  sphere,  and  also  the 
angle  which  the  arc  of  a  great  circle,  drawn  from 
one  to  the  other,  makes  with  the  meridian  passing 
through  either,  assumed  as  the  central  body  ;  from 
these  polar  co-ordinates,  the  apparent  orbit,  as  pro- 
iected  upon  the  celestial  sphere,  is  easily  traced. 

§  579.  The  relation  which  is  found  to  connect  the  distances  with  the 
angular  velocities  shows  the  stars  to  be  under  the  control  of  a  central 
force,  and  the  elliptical  form  of  the  orbit,  with  the  eccentric  position  of 
the  central  star,  is  proof  that  this  force  can  be  no  other  than  that  of  grav 
rtatic/n. 

§  580.  Thus,  the  same  principle  which,  under  the  influence  of  distance, 
directs  the  satellites  about  their  primaries,  and  the  primaries  about  our  sun, 
also  wheels  distant  suns  around  suns,  each,  perhaps,  carrying  with  it  its 
system  of  planets,  and  each  planet  a  group  of  satellites. 

§  581.  From  the  micro  metrical  measurements  above  referred  to,  and 
the  intervals  of  time  between  them,  the  elements  of  the  actual  stellar 
orbits  are  easily  computed.*  A  number  of  sets  are  given  in  the  following 
table : 


See  Memoirs  of  Royal  Astronomical  Scoiety,  voL  v.  p.  111. 


156 


SPHERICAL    ASTRONOMY. 


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STARS.  157 

§  582.  If  the  annual  parallax  of  the  system,  the  apparent  semi-axis 
of  the  stellar  orbits,  and  the  earth's  radius  vector,  be  substituted  respec- 
tively for  P,  s,  and  p,  in  Eq.  (29),  d  will  become  the  number  of  linear 
r.nits  in  the  mean  distance  between  the  stars. 

Assuming  the  data  of  the  table,  selecting  a  Centauri,  and  making 
*  =  15  ".5  and  P  =  0.913,  we  have 


whence  the  stellar  orbits  of  a  Centauri  are  (§  422)  about  nine-tenths  that 
cf  Uranus. 

§  583.  Denoting  by  T  the  periodic  time  of  a  body  about  its  centre,  we 
have,  Analyt.  Mechanics,  §  201, 

7*  -4**a' 

" 


in  which  a  is  the  mean  distance,  *  the  ratio  of  the  circumference  to  diam- 
eter, and  k  the  intensity  of  the  central  attraction  at  the  unit's  distance. 
For  a  second  body 


hence 


but  from  the  laws  of  gravitation  k  and  kr  are  directly  proportional  to  the 
attracting  masses,  and  we  have 


Making  a  =  p,  a'  =.  d  =  16.977  p  ;  T—  1  year,  and  T'~  77  years  ;  then 
will  M  denote  the  mass  of  the  sun  and  M'  that  of  the  central  star  of 
a  Centauri,  and  we  have  from  the  above  proportion 


JT=  .  JT=  0.88.  Jf; 

that  is,  the  mass  of  the  central  star  is  a  little  over  eight-tenths  that  of 
:  ur  sun. 

§  584.  Color  of  Double  Stars.  —  Many  of  the  double  stars  present  the 
curious  phenomena  of  complementary  colors.  In  such  instances  the  larger 
star  is  usually  of  a  ruddy  or  orange  hue,  while  the  smaller  one  appears 
blue  or  green.  The  double  star  i  Cancri  presents  the  beautiful  contrast  of 


158  SPHERICAL    ASTRONOMY. 

yellow  and  blue ;  y  Andromeda,  crimson  and  green.  Where  there  i8 
great  difference  in  the  magnitudes  of  the  individuals,  the  larger  is  usually 
white,  while  the  smaller  may  be  colored ;  thus,  i\  Cassiopeia;  exhibits  the 
beautiful  combination  of  a  large  white  star  and  a  small  one  of  a  rich  ruddy 
purple.  If  this  be  not  the  mere  optical  effect  of  contrast  of  brightness, 
what  variety  of  illumination  two  suns — a  red  and  a  blue  one,  a  crimson 
and  green  one — must  afford  to  the  inhabitants  of  planets  that  circulate 
around  them,  having  sometimes  both  suns  above  their  horizon  at  once  and 
at  others  each  in  succession,  thus  producing  an.  alternation  of  red  and  blue, 
crimson  and  green  days !  Insulated  stars  of  a  reJ  color,  almost  as  deep  as 
blood,  occur  in  many  parts  of  the  heavens. 

§  585.  Proper  Motions  of  the  Stars. — As  might  be  expected  from  their 
mutual  attractions,  however  enfeebled  by  distance  and  opposing  attractions 
from  opposite  quarters,  the  stars  are  found  to  have  a  proper  motion,  which 
in  the  lapse  of  time  has  produced  a  sensible  change  of  internal  arrange- 
ment. Thus,  from  the  time  of  Hipparchus,  130  years  B.  c.,  to  A.  D.  1717, 
eighteen  hundred  and  forty -seven  years,  the  conspicuous  stars  Sirius,  Arc- 
turus,  and  Aldobaran,  are  found  to  have  changed  their  latitudes  respect- 
ively 37',  42',  and  33',  in  a  southerly  direction.  Besides,  the  observations 
of  modern  astronomy  prove  that  such  motions  do  really  exist.  The  two 
stars  61  Cygni  are  found  to  have  retained  sensibly  unchanged  their  dis- 
tance apart  for  the  last  fifty  years,  while  they  have  shifted  their  places  in 
the  heavens  in  the  same  interval  no  less  than  4'  23",  giving  an  annual 
proper  motion  to  each  of  5 ".3.  Of  the  stars  not  double,  and  no  way  dif- 
fering from  the  rest  in  any  other  sensible  particular,  s  Indi  and  JUL  Cassio- 
peia? have  the  greatest  proper  motions,  -amounting  annually  to  7".74  and 
3 ".74  respectively. 

§  586.  Proper  Motion  of  the  Sun. — The  inevitable  consequence  of  a 
proper  motion  in  our  sun,  if  not  equally  participated  in  by  the  rest,  must 
be  a  slow  average  apparent  tendency  of  all  the  stars  to  the  point  of  the 
celestial  sphere  from  which  the  sun  is  moving,  and  a  corresponding  retro- 
cession from  the  opposite  point — and  this,  however  greatly  individual  star? 
may  differ  from  such  average  by  reason  of  their  own  peculiar  proper  mo- 
tion. This  is  the  necessary  effect  of  parallax,  and  has  been  detected  by 
observation. 

By  properly  treating  the  observations  on  the  stars  of  the  northern  hemi- 
sphere, the  solar  apex,  as  it  is  called,  or  the  point  towards  which  the  sun 
was  moving  at  the  epoch  of  1790,  was  in  right  ascension  250°  09',  and 
north  polar  distance  55°  23'.  The  southern  stars  gave,  by  a  similar  mode 
of  treatment,  right  ascension  260°  01',  and  n^rth  polar  distance  55°  37': 


Plate  IX. 


NEBULAE.  159 

results  so  nearly  identical  as  to  remove  all  doubt  of  the  sun's  proper 
motion. 

§  587.  All  analog}7  would  lead  to  the  conclusion  that  the  sun  is  de- 
scribing an  orbit  of  vast  extent  about  the  centre  of  gravity  of  the  group  of 
stars  of  which  it  forms  a  single  member,  and  of  which  the  milky  way  is 
to  us  but  the  distant  trace,  while  this  group  may  itself  be  moving  as  a 
single  system  around  some  other  and  vastly  distant  centre.  A  line  drawn 
tangent  to  the  solar  orbit  in  1790  pierced  the  celestial  sphere  near  the 
stars  if  Herculis  and  a  Columba,  the  sun  being  then  moving  towards  the 
former  and  from  the  latter.  And  the  result  of  calculations  thus  far  gives 
to  the  sun  a  velocity  of  422,000  miles  a  day,  or  little  more  than  one-fourth 
the  earth's  rate  of  annual  motion  in  its  orbit. 


NEBULAE. 

§  588.  Besides  the  stars  which  appear  as  shining  points,  there  are 
cloud-like  patches  of  light  to  be  seen  scattered  here  and  there  over  the 
celestial  vault.  These  are  called  nebulce.  They  present  themselves  under 
great  variety  of  shapes  and  sizes,  as  exemplified  in  Figs.  12,  13,  14  (front- 
ing plate),  and  exhibit  in  the  telescope  different  characters  of  internal 
structure  with  every  increase  of  optical  power.  They  are  very  unequally 
distributed  over  the  heavens.  In  the  northern  hemisphere,  the  hours  3,  4, 
5,  16,  17,  and  18  of  right  ascension  are  singularly  poor,  while  the  hours  10, 
11,  and  12,  especially  the  latter,  are  exceedingly  rich  in  these  objects.  In  the 
southern  hemisphere  a  much  greater  uniformity  prevails,  with  two  remark- 
able exceptions,  to  be  noticed  presently.  They  have  no  decided  tendency 
to  any  particular  region. 

§  589.  When  viewed  through  the  telescope,  many  nebulae  are  resolved 
into  stars,  and  the  number  that  thus  yield  their  cloud-like  aspect  increases 
with  every  augmentation  of  instrumental  power.  Nebulae  are  therefore 
classified,  with  reference  to  their  appearance  through  the  telescope,  into 
resolvable,  irresolvable,  planetary,  and  stellar  nebulce,  and  nebular  stars. 

§  590.  Resolvable  Nebulce. — These  are  usually  called  clusters  af  stars. 
Some  are  very  broken  in  outline,  while  others  are  so  regular  as  to  suggest 
the  prevalence  of  some  internal  action  productive  of  symmetrical  arrange- 
ment among  their  internal  parts. 

§  591.  Irregular  clusters  are  much  less  rich  in  stars,  and  much  less 
condensed  towards  the  centre.  In  some  the  stars  are  nearly  of  the  same 
size,  in  others  very  different.  The  group  called  the  Pleiades,  in  which  six 


160  SPHERICAL   ASTRONOMY. 

or  seven  stars  may  be  counted  with  the  naked  eye,  and  fifty  or  sixty  with 
the  telescope,  is  one  of  the  most  obvious  examples  of  this  class.  Coma 
Berenices,  represented  in  Fig.  15,  Plate  X,  is  another  such  group. 

§  592.  Globular  Clusters. — These  take  their  name  from  their  round 
appearance.  They  are  much  more  difficult  of  resolution,  and  some  have 
frequently  been  mistaken  for  comets  without  tails.  When  viewed  through 
the  telescope,  they  are  found  to  be  composed  of  stars  so  crowded  together 
as  to  occupy  an  almost  definite  outline,  and  to  run  up  to  a  blaze  of  light 
towards  the  centre,  where  their  condensation  is  greatest.  It  would  be  vain 
to  attempt  to  count  the  stars  in  these  clusters ;  some  have  been  estimated 
to  contain  five  thousand,  within  an  area  not  greater  than  the  tenth  part  of 
the  lunar  disk. 

§  593.  Elliptic  Nebulae. — The  figure  here  again  suggests  the  name. 
They  are  of  all  degrees  of  eccentricity,  from  moderately  oval  to  elongations 
so  great  as  to  *be  almost  linear.  In  all,  the  density  increases  towards  the 
centre,  and  generally  their  internal  strata  approach  more  nearly  the  spheri- 
cal form  than  their  external.  Their  resolvability  is  greater  in  the  central 
parts ;  in  some  the  condensation  is  slight  and  gradual,  in  others  great  and 
sudden. 

The  largest  and  finest  specimen  of  elliptic  nebulae  is  in  the  Girdle  of 
Andromeda,  given  in  Fig.  12,  Plate  IX. 

§  594.'  Annular  nebulae  also  exist,  but  are  very  rare.  The  most  con- 
spicuous of  this  class  is  found  between  /3  and  y  Lyrae,  and  may  be  seen 
through  a  telescope  of  moderate  power.  The  central  vacuity,  Fig.  16, 
Plate  IV,  is  not  quite  dark,  but  appeal's  as  a  light-colored  gauze  stretched 
over  a  hoop.  The  powerful  telescope  of  Lord  Rosse  resolves  this  nebula 
into  excessively  minute  stars,  and  shows  filaments  of  stars  hanging  to 
its  edge. 

§  595.  Spiral  Nebulae. — These  are  most  curious  objects.  Their  dis- 
covery is  but  very  recent,  and  is  due  to  the  powerful  instrument  of  Lord 
Rosse.  As  their  name  indicates,  they  appear  to  consist  of  a  spiral  or  vor- 
ticose arrangement  of  stars  diverging  from  a  centre,  and  suggest  the  idea 
of  a  vast  self-luminous  mass  of  matter,  travelling  to  a  common  destination 
along  separate  curvilinear  paths.  Their  form  and  general  appearance  are 
represented  in  Figs.  17  and  18,  Plate  X. 

§  596.  Planetary  Nebulce. — These  take  their  name  from  the  planet- 
like  disk  which  they  present.  In  some  instances  they  bear  a  perfect  re- 
semblance to  a  planet  in  this  respect,  being  round  or  slightly  oval,  and 
quite  sharply  terminated.  In  some  the  illumination  is  perfectly  equable: 
in  others  mottled,  and  of  a  peculiar  texture,  as  if  curdled.  They  are  com- 


Plato  X. 


TO 


m  ' 

(^UNIVERSITY 


162  SPHERICAL   ASTRONOMY. 

These  nebulae  may  be  grouped  into  four  great  masses,  which  occupy  the 
regions  of  Orion,  of  Argo,  of  Sagittarius,  and  of  Cygnus. 

§  601.  The  Magellanic  Clouds,  or  the  Nubeculce  (Major  and  Minor), 
as  they  are  called  in  celestial  maps  and  charts,  are  two  nebulous  or  cloudy 
masses  of  light  conspicuously  visible  to  the  naked  eye  in  the  southern 
hemisphere,  and  in  appearance  and  brightness  resemble  portions  of  the 
milky  way  of  the  same  size.  They  are  in  shape  somewhat  oval,  the  larger 
deviating  most  from  the  circular  form.  The  larger  is  situated  between  the 
hour  circles  4h  40m  and  6h  40m,  the  parallels*  156°  and  162°  north  polar 
distance,  and  occupies  an  area  of  about  42  square  degrees.  The  lesser, 
which  is  between  the  hour  circles  .Oh  28m  and  lh  15m,  and  the  parallels  of 
162°  and  165°  north  polar  distance,  covers  about  ten  square  degrees.  The 
general  ground  of  both  consists  of  large  tracts  of  nebulosity  in  every  stage 
of  resolubility,  from  light  irresolvable  up  to  perfectly  separated  stars  like 
the  milky  way,  including  groups  sufficiently  insulated  and  condensed  tc 
come  under  the  designation  of  irregular  and  globular  clusters,  the  latter 
being  in  every  stage  of  condensation.  In  addition  they  contain  nebular 
objects  quite  peculiar,  and  which  have  no  analogy  in  any  other  part  of  the 
heavens.  Globular  clusters,  except  in  one  region  of  small  extent,  and  neb- 
ulye  of  regular  elliptic  forms  are  comparatively  rare  in  the  milky  way,  but 
are  congregated  in  greatest  abundance  in  parts  of  the  heavens  the  most  re- 
mote possible  from  the  gallactic  circle ;  whereas  in  the  Magellanic  Clouds 
they  are  indiscriminately  mixed  with  the  general  starry  ground. 

§  602.  Regarding  the  nubeculce  as  spherical  in  form,  and  not  as  vastly 
long  vistas  foreshortened  by  having  their  ends  turned  towards  the  earth — 
which  would  be  improbable  seeing  there  are  two  of  them  close  together — 
the  brightness  of  objects  in  their  nearer  portions  cannot  be  much  exagger- 
ated, nor  those  in  its  remoter  much  enfeebled  by  difference  of  distance. 
It  must,  therefore,  be  an  admitted  fact  that  stars  of  the  7th  and  8th  mag- 
nitudes and  irresolvable  nebulae  may  coexist  within  limits  of  distance  com- 
paratively small,  and  that  all  inferences  in  regard  to  relative  distance 
drawn  from  relative  magnitudes  must  be  received  with  caution. 

§  603.  Our  Sun  a  Nebulous  Star. — Various  phenomena  indicate  that 
our  sun  is  itself  a  nebulous  star.  The  chief  is  that  called  the  zodiacal 
light,  which  may  be  seen  on  any  clear  evening  soon  after  sunset  about  the 
months  of  March,  April,  and  May,  and  at  the  opposite  seasons  of  the  year 
just  before  sunrise,  as  a  conically-shaped  light,  extending  from  the  hori- 
zon upwards  in  the  direction  of  the  sun's  equator.  The  apparent  angular 
distance  of  its  vertex  Ffrom  the  sun  S  varies  from  40°  to  90°,  and  its 
breadth  at  its  base,  perpendicularly  to  its  lei  gth,  firm  8°  to  30°.  Every 


Plate  Xtt. 


TO  FRONT  &AGE  163  . 


NEBULA.  lf>3- 

circumstance  connected  with  it  indicates  it  to  be  F'^-  **• 

a  lenticularly-formed   envelope  surrounding  the 

sun,  and  extending  beyond  the  orbits  of  Mercury     Ffrrizcii  ^ 

and  Venus  and  even  to  the  Earth,  its  vertex 
having  been  seen  90°  from  the  sun  in  a  great 
circle.  Different  parts  of  the  heavens  furnish 
examples  of  similar  forms.  Figs.  25,  26,  27, 
Plate  XIT. 

§  604.  Aerolites. — Nothing  prevents  that  the  particles  of  this  vast  ma- 
terial envelope  may  have  tangible  size  and  be  at  great  distances  apart,  and 
yet  compared  with  the  planets,  so  called,  be  but  as  dust  floating  in  the 
sunbeam.  It  is  an  established  fact  that  masses  of  stone  and  lumps  ol 
iron,  called  Aerolites,  do  occasionally  fall  upon  the  earth  fiom  the  upper 
regions  of  the  atmosphere,  and  that  they  have  done  so  since  the  earliest 
records.  On  the  26th  April,  1803,  one  of  these  bodies  fell  in  the  imme- 
diate vicinity  of  the  town  of  L'Aigle,  in  Normandy,  and  by  its  explosion 
into  fragments,  scattered  thousands  of  stones  over  an  area  of  thirty  square 
miles.  Four  instances  are  recorded  of  persons  having  been  killed  by  the 
descent  of  such  bodies,  and  after  every  vain  attempt  to  account  for  them 
as  coming  originally  from  the  earth,  and  even  from  the  moon,  by  volcanio 
projections,  their  planetary  nature  is  now  generally  admitted.  Their  heat 
when  fallen,  the  igneous  phenomena  which  accompany  them,  their  explo- 
sion on  reaching  the  denber  regions  of  our  atmosphere,  are  accounted  for 
by  the  condensation  in  front  of  them  created  by  their  enormous  velocity, 
and  bty-  the  relations  of  air,  in  a  highly  attenuated  state,  to  heat. 

§  605.  Meteors. — Besides  these  more  solid  bodies,  others  of  much  less 
density  appear  also  to  be  circulating  around  the  sun  at  the  distance  of  the 
earth  from  that  luminary.  These  on  corning  within  the  atmosphere  ap- 
pear as  shooting  stars,  followed  by  trains  of  light,  '\nd  are  called  Meteors. 
They  appear  now  and  then  as  great  fiery  balls,  traversing  the  upper  re- 
gions of  the  atmosphere,  sometimes  leaving  long  luminous  trains  behind 
them,  sometimes  bursting  with  a  loud  explosion,  and  sometimes  becoming 
quietly  extinct.  Among  these  latter  may  be  mentioned  the  remarkable 
meteor  of  August  18th,  1783,  which  traversed  the  whole  of  Europe,  from 
Shetland  to  Rome,  with  a  velocity  of  30  miles  a  second,  at  a  height  of  50 
miles  above  the  earth,  with  a  light  greatly  surpassing  that  of  a  full  moon, 
and  diameter  quite  half  a  mile.  It  changed  its  form  visibly  and  quietly, 
separated  into  several  distinct  pprts,  which  proceeded  in  parallel  direc- 
tions, each  followed  by  a  train. 

§  606,    On    several   occasions    meteors    have   appeared   'n  «*tonishin£ 


16J.  SPHERICAL    ASTRONOMY. 

cumbers,  falling  like  a  shower  of  rockets  or  flakes  of  SIKW,  illuminating 
at  once  whole  continents  and  oceans,  even  in  both  hemispheres.  And  it 
is  significant  that  these  displays  have  occurred  between  the  12th  and  14th 
November  and  9th  and  llth  August.  In  November  they  are  much  more 
brilliant,  but  their  returns  less  certain  than  in  August,  when  numerous 
large  and  brilliant  shooting-stars  with  trains  are  almost  sure  to  be  seen. 

§  607.  Annual  periodicity,  irrespective  of  geographical  location,  points 
at  once  to  the  place  of  the  earth  in  its  orbit  as  a  necessary  concomitant, 
and  leads  to  the  conclusion  that  at  that  place  the  earth  enters  a  stratum, 
or  annular  stream  of  meteoric  planets,  in  their  progress  of  circulation 
around  the  sun.  The  earth  plunging  in  its  annual  course  into  a  ring  of 
these  bodies,  and  of  such  thickness  as  to  be  traversed  in  a  day  or  two,  their 
motions,  referred  to  the  earth  as  at  rest,  would  be  sensibly  uniform,  recti- 
linear, and  parallel.  Viewed  from  the  centre  of  the  earth,  or  from  any 
point  on  its  surface,  neglecting  the  diurnal  as  being  insignificant  in  com- 
parison with  the  annual  motion,  their  paths  wojild  appear  to  diverge  from 
a  common  point  on  the  celestial  sphere.  Now  this  is  precisely  what  haj>- 
pens.  The  vast  majority  of  the  November  meteors  appear  to  describe  arcs 
of  great  circles  passing  through  y  Leonis,  and  those  of  August  appear 
to  move  along  paths  having  a  common  point  in  /3  Camelopardi. 

§  608  As  the  ring  may  have  any  position  and  be  of  an  elliptical  fig- 
ure having  any  reasonable  eccentricity,  both  the  velocity  and  direction  of 
«<ac.h  meteor  may  differ  to  any  extent  from  those  of  the  earth,  so  there  is 
nothing  in  the  great  difference  of  latitude  of  these  meteoric  apices  at  all 
opposed  to  the  foregoing  conclusion. 

§  609.  If  the  meteoric  planets  were  uniformly  distributed  in  the  sup- 
posed ring,  the  earth's  annual  encounter  with  them  would  be  certain  if  it 
occurred  once ;  but  if  such  ring  be  broken,  and  the  bodies  revolve  in 
groups,  with  periods  differing  from  that  of  the  earth,  years  may  pass  with- 
out rencontre,  and  when  such  happen,  they  may  differ  to  any  extent  in 
intensity  of  character,  according  as  the  groups  encountered  are  richer  or 
poorer  in  the  number  of  their  elements. 

§  610.  From  careful  observations,  made  at  the  extremities  of  a  base 
50,000  feet  long,  it  has  been  inferred  that  the  heights  of  meteors  at  the 
instant  of  first  appearance  and  disappearance,  vary  from  16  to  140  miles, 
and  their  relative  velocities  from  18  to  36  miles  a  second.  Altitudes 
and  velocities  so  great  as  these  clearly  indicate  an  independent  planetary 
circulation  round  the  sun. 

§  611.  It  is  not  impossible  that  some  of  these  bodies  may  have  been 
converted  by  the  superior  attraction  of  the  earth,  arising  from  greater  prox- 


EPHEMERIDES.  1(J^ 

imity,  into  permanent  satellites ;  and  there  are  those  who  believe  in  the 
existence  of  at  least  one  of  these  bodies,  which  completes  its  circuit  about 
the  earth  in  about  3h  20m,  and  therefore  at  a  mean  distance  of  about 
5000  miles. 

EPHEMERIDES. 

§  612.  The  facts  and  principles  now  explained  enable  us  to  predict  the 
aspect  of  the  heavens,  or  positions  of  the  heavenly  bodies,  for  all  future 
time.  This  prediction  is  usually  drawn  up  in  the  condensed  form  of  tables, 
which  are  called  ephemerides.  The  table  relating  to  any  one  body  is  called 
the  ephemeris  of  that  body,  as  the  ephemeria  of  the  sun,  of  the  moon,  <fee. 

§  613.  Ephemerides  are  prepared  in  advance  to  subserve  the  wants  and 
promote  the  interests  of  navigation,  geography,  and  chronology,  as  well  as 
of  future  astronomical  discovery  and  research. 

§  614.  To  facilitate  the  computation  of  the  ephemeris  of  a  body,  it  ie 
usual  first  to  construct  what  are  called  its  tables  ;  and  the  manner  of  doing 
this  may  best  be  explained  by  taking  a  particular  example,  say  that  of  the 
sun,  or  rather  the  earth,  since  this  is  the  moving  body ;  but  as  the  place  of 
the  sun,  as  seen  from  the  earth,  differs  from  that  of  the  earth  as  seen  from 
the  sun  by  the  constant  180°,  we  shall  speak  of  the  sun. 

§  615.  We  have  seen,  §  197,  how  the  mean  longitude  of  the  sun,  his 
mean  motion,  longitude  of  the  perigee,  and  eccentricity,  may  be  found 
from  observation  and  computation.  These  elements  being  found  at  epochs 
widely  separated  from  one  another,  the  changes  which  take  place  in  the 
last  three,  and  the  rate  of  motion  of  the  perigee,  are  ascertained. 

§  616.  Having  fixed  upon  any  epoch,  say  mean  noon  or  midnight,  1st 
January,  1800,  any  interval  of  time,  either  after  or  before  the  epoch,  mul- 
tiplied by  the  mean  motion  of  the  sun  in  longitude,  will  give  the  increase 
of  mean  longitude  during  that  interval,  and  being  added  to  the  mean  lon- 
gitude at  the  epoch  and  the  sum  divided  by  360°,  the  remainder  will  give 
the  mean  longitude  at  the  beginning  of  the  interval,  if  it  be  before,  or  end, 
if  it  be  after  the  epoch.  These  longitudes,  with  the  corresponding  dates, 
being  tabulated,  give  what  is  called  a  table  of  epochs,  which  tells  by  simple 
inspection  the  mean  longitude  on  any  given  day,  hour,  minute^  and  second. 

§  617.  The  same  process  being  performed  with  reference  to  the  longi- 
tude of  the  perigee  and  its  rate  of  change,  gives  a  corresponding  table  in 
which  the  longitude  of  the  perigee  is  found. 

§  618.  Resuming  Eq.  (o),  Appendix  No.  V.,  and  causing  m  t' ,  which 
is  the  mean  anomaly,  to  vary  from  0°  to  360°,  correspond  w  equations  of 


SPHERICiL  ASTRONOMY. 

the  centre  will  result,  and  these  properly  arranged  form  a  table  of  equation* 
of  the  centre,  of  which  the  arguments,  as  they  are  called,  are  the  mean 
anomalies.  Then  causing  the  eccentricity  to  vary  according  to  ascertained 
rates,  the  same  equation  gives  the  elements  of  an  additional  table  by  which 
the  equations  of  the  centre  may  be  corrected  from  time  to  time. 

§  019.  Nutation  causes  the  true  equinox  to  oscillate  about  a  mean 
place,  its  distance  therefrom  being  equal  to  the  algebraic  sum  of  two  func- 
tions, of  which  one  depends  upon  the  longitude  of  the  moon's  node,  the 
other  upon  the  longitude  of  the  sun,  and  both  upon  the  obliquity  of  the 
ecliptic.  Tables  containing  the  values  of  these  functions  for  assumed  places 
of  the  moon's  node  and  of  the  sun,  give  the  numbers  whose  sum  is  equal 
to  the  equation  of  the  equinoxes  in  longitude. 

§  620.  In  addition,  the  larger  of  the  planets,  especially  Venus  and  Ju- 
piter, disturb  the  earth's  orbit.  These  perturbations  are  computed  by  pro- 
Cesses  in  physical  astronomy,  and  their  values  arranged  under  heads  that 
give  the  angular  distances  of  the  disturbing  planets  from  the  earth  as  seen 
from  the  sun,  and,  together  with  the  place  of  the  moon's  node,  furnish  the 
argument  v7Ub  which  other  tables  are  entered  that  give  the  corresponding 
effects  upo*i  the  sun's  longitude. 

§  621.  Lastly,  as  the  purpose  is  to  find  the  place  where  the  sun's  centre 
is  to  be  seen,  provision  is  made  for  the  effect  of  aberration.  This  in 
ihe  case  of  the  sun  is  nearly  constant,  and  equal  to  —  20".25,  because 
of  the  small  eccentricity  of  the  earth's  orbit,  the  greatest  variation  there- 
from being  less  then  0".35.  This  constant  is  included  in  the  epoch  tables. 

§  622.  Epkemeris  of  the  Sun. — We  are  now  prepared  to  find  where  the 
sun  has  been  and  where  he  will  be  on  the  celestial  sphere  throughout  time. 
For  this  purpose,  enter  the  table  of  epochs  with  the  date,  take  out  his  mean 
longitude  and  the  longitude  of  the  perigee  ;  the  difference  will  be  the  mean 
anomaly,  with  which  enter  the  table  of  the  equations  of  the  centre  and 
take  out  the  corresponding  equation ;  add  this  to  or  subtract  it  from  the 
mean  longitude  according  to  its  sign,  and  the  result  will  be  the  true  lon- 
gitude of  the  sun  as  affected  by  nutation  and  perturbations.  Take  these 
latter  from  the  appropriate  tables,  and  we  have 

True  longitude  of  sun  =  mean  longitude  -\-  equation  of  the  centre  -f- 
nutation  or  equation  of  equinoxes  in  longitude  -{-perturbations. 

§  623.  With  the  true  longitude  and  obliquity  of  the  ecliptic,  we  pass, 
by  spherical  trigonometry,  §  149,  to  right  ascension  and  declination. 

§  624.  The  mean  anomaly  in  Eq.  (»),  Appendix  V.,  gives  the  corres- 
ponding true  anomaly ;  and  the  latter  in  Eq.  (c),  same  Appendix,  gives  the 


EPHEMERIDES.  167 

••• 

radius  vector  »•,  which  in  equations  (28)  and  (29)  give  the  correspond- 
ing horizontal  parallax  and  apparent  diameter. 

§  625.  The  mean  longitude  corrected  for  the  equation  of  the  equi- 
noxes in  right  ascension,  and  diminished  by  the  right  ascension,  gives 
the  equation  of  time. 

§  626.  These  and  other  elements  being  determined  at  different  epochs, 
say  for  every  noon  on  some  fixed  meridian,  their  consecutive  differences, 
divided  by  the  number  of  hours  between  the  epochs,  give  the  hourly 
changes,  and  therefore  the  means  of  finding  the  value  of  the  elements 
themselves  for  any  other  meridian. 

The  elements  with  their  hourly  changes  make  up,  when  properly  tabu- 
lated, an  ephemeris  of  the  sun. 

§  627.  Ephemeris  of  the  Moon. — The  motion  of  the  moon  is  altogether 
more  irregular  and  complicated  than  the  apparent  motion  of  the  sun,  owing 
mainly  to  the  disturbing  action  of  this  latter  body.  But  these  and  other 
perturbations  have  been  computed  and  tabulated,  and  from  these  tables, 
including  those  of  the  node  and  inclination,  the  places  of  the  moon  in  her 
orbit  are  found  in  much  the  same  way  as  those  of  the  sun  in  the  ecliptic. 
The  mean  orbit  longitude  of  the  moon  and  of  her  perigee  are  first  found 
and  corrected :  their  difference  gives  her  mean  anomaly,  opposite  to  which 
in  the  appropriate  table  is  found  the  equation  of  the  centre,  and  this  being 
applied  with  its  proper  sign  to  the  mean  orbit  longitude  gives  the  true 
orbit  longitude. 

§  628.  Let  E  be  the  earth,  M.the  moon,  FJs-  "• 

V  the  vernal  equinox,  VM'  an  arc  of  the 
ecliptic,  VQ  of  the  equinoctial,  and  JOf'of  a 
circle  of  latitude;  then  will  MM'  be  the 
latitude  and  VM'  the  longitude  of  the  moon, 
VN  the  longitude  of  the  node  and  VEN 
+  N EM  the  orbit  longitude  of  the  moon. 

Subtracting  from  the  orbit  longitude  of 
the  moon  the  longitude  of  the  node,  the  re- 
mainder NM  will  be  the  moon's  angular  distance  from  her  node.  This 
and  the  inclination  M N M'  will  give,  in  the  right-angled  triangle  M  N  M\ 
the  latitude  MM'  and  the  side  NM',  which  latter  added  to  the  longitude 
of  the  node  N  V  gives  the  longitude  V M' .  The  latitude  and  longitude, 
together  with  the  obliquity  of  the  ecliptic,  give,  §  153,  the  right  ascension 
and  declination.  The  radius  vector,  equatorial  horizontal  parallax,  apparent 
diameter,  &c.,  are  computed  as  in  the  case  of  the  sun.  And  thus  an 
ephemeris  of  the  moon  is  constructed. 


168 


SPHERICAL    ASTRONOMY. 


Fig.  100. 


§  b*'29.  Ephemeris  of  a  Planet. — From  tables  of  a  planet  its  true  orbit 
longitude  as  seen  from  the  sun  is  found,  as  in  the  case  of  the  moon  as  seen 
from  the  earth.  From  the  heliocentric  orbit  longitude,  heliocentric  longi- 
tude of  the  node,  and  inclination,  the  heliocentric  longitude  and  latitude, 
together  with  the  radius  vector,  are  found ;  just  as  the  corresponding  geo- 
centric elements  of  the  moon  are  found  from  similar  data  relating  to  the 
lunar  orbit ;  and  from  the  heliocen- 
tric longitude,  latitude,  and  radius 
vector,  we  pass  to  the  geocentric, 
thus :  . 

§  630.  Let  P  be  the  planet,  E 
the  earth,  S  the  sun,  and  0  the 
projection  of  the  planet  upon  the 
plane  of  the  ecliptic.  Draw  from 
S  and  E  the  parallels  S  V  and 
EV  to  the  vernal  equinox,  and 
make 

r  =  JES       =  radius  vector  of  earth ; 

r'  =  SP       =  radius  vector  of  planet; 

X  =  V S  0   —  heliocentric  longitude  of  planet; 

X'  =  V  E  0  =  geocentric  longitude  of  planet ; 

&  =  P  S  0  =  heliocentric  latitude  of  planet ; 

&'  =  P  E  0  =  geocentric  latitude  of  planet ; 

S  =  0  S  E  =  commutation ; 

0  =  S  0  E  =  heliocentric  parallax  ; 

E  =  SE  0  =  elongation ; 

0  =  V  ES  =  longitude  of  sun. 


Then 
and  because 


S  0  =  r'  cos  6  ; 


VST= 


=360°-  0 


we  have 


S=  180°— (360°—  0)  —  X=  ©  -  180° -X; 


whence  the  commutation  is  known.     Then  in  the  plane  triangle  OES, 
r'  cos  d  +  r  :  r'  cos  6  —  r  :  :  tan  £  (E  +  0)  :  tan  £  (E  —  0) ; 


but 
whence 


S+ 


=  180°, 

90°-- 

""  2 


(163) 


EPHEMEKIDES.  169 

Substituting  this  above,  we1  have 

tan  i  (E-  0)  =  cot  $S  .  ^A^; 
rf  cos  6  +  r  ' 


and  making 

r'  cos 


tan  J(^-  0)  =  cot  J  £  .  ta"  X  ~—  =  cot^.tanfa  -  45°)  .  .  (164) 
tan  }£  -f-  1 

Knowing  from  Eq.  (163)  the  half  sum  of  E  and  0,  and  from  Eq.  (164) 
their  half  difference,  E  and  0  become  known. 
And  we  have 

\'  =  E  -  (360°  -  0)  =  E  +  O  -  360°  .     .     .     (165) 

§  631.  Again; 

P  0  =  E  0  .  tan  0'  =  S  0  .  tan  6  ; 
whence 

tend'       SO       s\u£! 


tan  6   ~~    £0  ~~  sin  S  ' 
and 

tan  6'  =  tan  4  .  ^-^     .     .  (166) 

sin  S 

From  equations  (165)  and  (166)  the  geocentric  longitude  and  latitude  be- 
come known. 

§  632.  Denote  by  r"  the  distance  EP  of  the  planet  from  the  earth  ; 
then  will 

E  0  =  r"  cos  V  and  S  0  =  r'cos  6  ; 

and  in  the  triangle  E  S  0 

r"  cos  &'  :  r'cos  6  :  :  sin  S  :  sin  E\ 
whence 

r,,  =  r,  cos*     sinS     ...... 

cos  0'    sin  ^ 

The  right  ascension,  declination,  horizontal  parallax,  and  apparent  diam- 
eter, are  found  as  in  the  case  of  the  sun  and  moon. 

§  633.  The  ephemerides  most  commonly  used  in  this  country  are  those 
computed  for  the  meridian  of  Greenwich,  England,  and  published  several 
years  in  advance  under  the  title,  "NAUTICAL  ALMANAC  AND  ASTRONOM- 
ICAL EPHEMERIS." 


170  SPHERICAL    ASTRONOMY. 


CATALOGUE  OF  STARS. 

§  634.  Another  important,  indeed  indispensable  auxiliary  to  practica. 
astronomy,  is  a  catalogue  of  stars.  This  consists  of  a  list  of  certain  stars 
arranged  in  the  order  of  their  right  ascensions,  with  the  means  of  obtaining 
the  right  ascensions  and  declinations  of  the  places  in  which  they  appear  at 
any  given  epoch. 

§  G35.  By  precession,  §  157,  nutation,  §  156,  and  aberration,  §  215. 
the  right  ascension  and  declination  of  a  star  are  ever  varying. 

The  place  of  a  star  referred  to  the  mean  equinoctial  and  mean  equinox 
is  called  its  mean  place;  that  referred  to  the  true  equinoctial  and  true 
equinox,  its  true  place ;  and  that  in  which  it  is  seen  referred  to  the  true 
equinoctial  and  true  equinox,  its  apparent  place. 

The  true  place  is  equal  to  the  mean,  corrected  for  nutation ;  and  the 
apparent  place  is  equal  to  the  true,  corrected  for  aberration.  The  true  and 
mean  places  are  found  from  the  apparent,  by  applying  the  same  correc- 
tions, with  their  signs  changed. 

§  636.  The  apparent  places  of  the  stars  are  used  as  points  of  reference 
on  the  celestial  sphere ;  and  knowing  the  right  ascensions  and  declinations 
of  these  places,  those  of  the  apparent  place  of  any  other  object  become 
known  also  when  the  distance  of  the  latter  in  right  ascension  and  declina- 
tion from  one  or  more  stars  is  found  by  instrumental  measurement. 

§  637.  Annual  Precession. — The  annual  precession  for  any  year  is, 

Luni-solar  =  50".3Y572  —  y  X  0".0002435890, 
General      =  50".21129  +  y  X  0".0002442966 ; 

in  which  y  denotes  the  number  of  the  year  from  1750,  minus  when  before 
that  epoch. 

§  688.  The  epoch  of  the  catalogue  which  will  be  referred  to  hereafter, 
that  of  the  British  Association,  is  January  1st,  1850.  Making  y  =  100, 
and  denoting  the  nutation  of  obliquity  by  4  w,  we  have 

A  «  =  9".2500  cos  &  -  0\0903  cos  2  ft  +  0".0900  cos  2  D  -f  0".5447  cos  2  ©  ; 

in  which  &>  denotes  the  mean  longitude  of  the  moon's  node,  D  the  true 
longitude  of  the  moon,  and  0  the  longitude  of  the  sun. 

§  639.  And  assuming  the  mean  obliquity  of  the  ecliptic  for  1850  equal 
to  w  =  23°  27'  31 ",  we  have  then  for  the  nutation  in  longitude,  denoted 
by  J  //, 

A  L—  -  17".3017  sin  &  +  0".20S1  sin  2  &  -  0".2074  sin  2  D  —  1".2552  sin  2  0 


CATALOGUE    OF    STARS  J^J 

£  640.  Denoting  the  equation  of  the  equinoxes  in  right  ascension  by 
J  A,  we  have 
A  A  =  —  15". 872  sin  ft  -f  0".192  S.TI  2  ft  —  0".190  sin  2  })  —  1".500  sin  2  0. 

§  641.  Denoting  the  right  ascension  and  declination  of  any  body  by 
a  and  5  respectively,  and  by  p  and  p',  its  change  in  the  same  due  to  an- 
nual precession,  then  will 

p  =  46".05910-f  20".05472  sin  a  .tan  d  .     .     .     (168) 
p'  =  20".05472  cos  a (169) 

§  642.  The  change  in  right  ascension  and  declination  for  any  fractional 
portion  of  the  year  will  be  found  by  multiplying  the  above  by 

'  =  365^25  =  °-002'73785  X  d      '     '     '     '     (170> 

In  which  d  denotes  the  number  of  days  from  the  beginning  of  the  year  to 
the  end  of  the  fraction. 

§  643.  Denoting  by  da.,  and  d8t  the  change  in  right  ascension  and  dec- 
lination arising  from  nutation,  then,  omitting  terms  involving  sin  2  D ,  will 

L-.  a,  =  —  (15".872  +  6".888  sin  a .  tan  i) .  sin  ft  —  '9".250  cos  a  .  tan  (5 .  cos  ft  ) 

•f  (o".191-f  0".083  sin  a  .  tan  I) .  sin  2ft+.0".090  cos  a .  tan  6.  cos  2ft  V  (171) 
—  (1".151+0".SOO  .  sin  a  .  tan  t) .  sin  2©  —  0".545  cos  a .  tan  S .  cos  2©  \ 

A  d,  =  9". 250  .  sin  a  .  cos  Q  —  6 ".888  cos  a  .  sin  ft       J 

-  0".090  sin  a  cos  2  ft  +  0".083  cos  a  .  sin  2  ft     [     .  (1Y2) 
+  0".545  sin  a  .  cos  2  0  —  0".500  cos  a  .  sin  2  O  5 

g  644.  Aberration. — Denoting  by  ^«8  and  4  6a  the  change  in  right 
ascension  and  declination  arising  from  aberration,  disregarding  the  eccen- 
tricity of  the  earth's  orbit, 

A  a2  s-s  -  (20".4200  sin  ©  .  sin  a  -f  18".7322  cos  Q  cos  a) .  sec  I  .     .     (178) 

A<52=  —  (20'  .4200  sin  Q  .  cos  a  -  18".7322  cos  O  sin  a)  sin  i  \ 
—  8".1289  cos  0  cos  i  ) 

§  645.  Multiplying  Eq.  (168)  by  Eq.  (170),  adding  together  the  prod- 
uct and  equations  (171)  and  (173),  and  denoting  the  apparent  right  as- 
cension by  a  and  the  mean  by  a',  there  will  result,  after  suitable  reduction, 

a'  —  a  =  Aa  =  («  —  0.848  rfn  fa  +  0.004  sin  2  ^  —  0.026  sia  2 0 )  X  (46".069  -f  20".OK>  sin  a  tan  to 
-  (9".260  cos  &  -  0''.090  cos  2  ft  +  0".545  cos  2  0 ) .  cos  a  .  tan  i 

—  20".420  sin  0  .  sin  a  .  sec  i 

—  18".732  cos  Q  .  cos  a  .  sec  6 

••   0".0530  sin  ft  -f  0".000  sin  2  ft  —  0".0039  sin  2  Q. 


SPHERICAL    ASTRONOMY. 

Mu/tiplying  Eq.  (169)  by  Eq.  (170),  adding  together  the  product  and 
equations  (172)  and  (174),  and  denoting  the  apparent  declination  by  d  and 
the  mean  by  <T,  we  also  have,  after  reduction, 

S'  -t—  &6  =  (t-  0.343  sin  Q,  -f  0.004  sin  2  ^  -  0.025  sin  2  Q)X20''.055  cos  a 
-f  (9".250  cos  Q  -  0".090  cos  2  Q  -f  0".545  cos  2  ©)  sin  a 

—  20".420  sin  Q  .  cos  a  .  tan  6 

—  18".732  cos  O  (tan  u  .  cos  6  —  sin  o  .  sin  3). 

Neglecting  the  three  last  terms  in  the  value  for  A  a  as  insignificant,  and 
making 

A=  —  18".732  cos  o, 

J5=  —  20".420  sin  O, 

C=t  —  0.025  sin  2  O  —  0.343  sin  &  +  0.004  sin  2  ft, 

D  =  —  0".545  cos  2  ©  —  9".250  cos  ft  +  0".090  cos  2  ft> 

a  =  cos  a  .  sec  £, 

b  =  sin  a  .  sec  £, 

c  —  46".059  +  20".055  sin  a  .  tan  6, 

d  •=  cos  a  tan  £, 

a'  =  tan  w  .  cos  §  —  sin  a  .  sin  5, 
&'  =  cos  a  .  tan  £, 
c'  =  20".055  cos  a, 
d'  =  —  sin  a ; 

the  above  become 

d.£     ....     (173) 
d'.D     ....     (176) 

§  646.  Proper  Motion. — To  the  foregoing  must  be  added  the  proper 
motion  of  the  star  when  it  is  known  with  sufficient  accuracy,  and  is  of 
sufficient  magnitude  to  be  taken  into  the  account. 

Equations  (173)  and  (174)  enable  us  to  pass  from  the  apparent  to  the 
true,  or  from  the  true  to  the  apparent  right  ascension  and  declination  of  a 
star. 

§  647.  Since  the  motion  of  the  equinoxes  is  very  slow,  the  values  of  the 
functions  a,  6,  c,  d,  a',  b',  c',  and  d'  will  be  sensibly  constant  for  a  number 
of  years,  particularly  when  the  stars  are  not  very  near  the  poles,  while 
those  of  the  functions  J,  B,  C,  and  D  vary  sensibly  from  day  to  day 
These  latter  are,  therefore,  computed  for  every  day  in  the  year,  and  their 
logarithms  recorded  in  the  astronomical  ephemeris ;  the  others  are  com- 
puted for  the  epoch  of  the  catalogue,  and  their  logarithms  recorded  oppo 
site  each  star  in  tbe  catalogue. 


CATALOGUE   Of  STARS. 


173 


g  64  g.  Construction  of  the  Catalogue. — The  elements  relating  to  each 
star  occupy  a  portion  of  the  two  pages  exposed  to  view  on  opening  the 
catalogue.  On  the  left-hand  page  will  be  found  every  thing  relating  to 
right  ascension,  and  on  the  right,  to  declination.  The  left-hand  page  con- 
sists of  eleven  vertical  columns :  in  the  first  is  placed  the  number  of  the 
star,  in  the  order  of  its  right  ascension ;  in  the  second,  the  name  of  the  con- 
stellation in  which  it  is  situated,  with  its  letter  or  number;  in  the  third, 
its  magnitude;  in  the  fourth,  its  mean  right  ascension,  January  1st,  1850, 
in  time ;  in  the  fifth,  its  mean  annual  precession  in  right  ascension,  Eq. 
(168),  reduced  to  time  ;  in  the  sixth,  its  secular  variation,  reduced  to  time ; 
in  the  seventh,  its  proper  motion  in  right  ascension,  reduced  to  time;  and 
in  the  eighth,  ninth,  tenth,  and  eleventh,  the  logarithms  of  the  functions 
a,  6,  c,  and  d,  reduced  to  time,  respectively,  each  preceded  by  the  sign  of 
the  function  to  which  it  belongs.  The  right-hand  page  consists  of  fifteen 
vertical  columns,  in  the  first  of  which  the  number  of  the  star  is  repeated ; 
the  second  contains  the  mean  north  polar  distance,  January  1st,  1850; 
the  third,  fourth,  and  fifth,  the  annual  precession,  secular  variation,  and 
proper  motion  in  north  polar  distance,  respectively ;  the  sixth,  seventh, 
eighth,  and  ninth,  the  logarithms  of  the  functions  a',  &',  c\  and  d'  re- 
spectively, each  preceded  by  the  sign  of  the  function  to  which  it  be- 
longs ;  the  remaining  columns  contain  the  numbers  by  which  the  star  is 
recognized  in  the  catalogues  of  the  several  authors,  whose  names  are  at 
the  top. 

Example. — Required  the  apparent  right  ascension  and  declination  oi 
y  Orionis,  February  5th,  1854. 


Mean  a  January  1st,  1850     . 
4  years'  prec.  and  pr.  motion 

Mean  a  January  1st,  1854     . 


b.  m.      ».  o      •       " 

5  17  05.33     Mean  N.  P.  D.    .     .     .          83  47  25.7 
-f-  1 2.88     4  y'ra'  prec.  and  pr.  motion  —  14.9 


5  17  18.21     Mean  N.  P.  D.   .     .     .    .     83  47  10.8 


a 
A 

a  A 

b 
B 

bB 


Logs. 

-f  8.0963 
—  1.1363 

-  9.2326 


-f  8.8188 
-f  1.1443 


9.9631 


Nat  No». 


-OM71 


-f  0.919 


A 
of  A 

9 

B 
VB 


Nat.  No* 


-f-  0.6483     .     -f-  4'  .449 

—  8.3039 
+  1.1443 


-  9.4482 


-  0.281 


174  SPHERICAL  ASTRONOMY. 

Logs.  Nat.  Nos.  Logs.  Nat 

c     .      -f  0.5070  c'     .     —  0.5721 

C          —  9.2812  C          —  9.2812 


e  C     .     -  9.7882  -  0.614         c'  C     .     +  9.8533     .      -f  0.713 


d     .     +  7.1304  d'     .     +  9.9923 

D     .     —0.5713  I>     .     —0.5713 

—  7.7017  —  0.005        d'D     .     —  0.5636     .     —  3.661 

A  a  =+0.129  AN.  P.  D.  =  +  1.220 

Hence  app't,  right  ascensk n,  Feb.  5,  1854,    5h  I7m  18-.21  -f  08.13  =    5h  17ra  188.34 
app't  N.  P.  D 83°  47'    10".80-f  1".22  =  83°  47'    12".02 


APPLICATIONS. 

TIME  OF  CONJUNCTION  AND  OF  OPPOSITION. 

§  649.  To  find  from  the  ephemeris  the  time  at  which  two  bodies  are  in 
conjunction  or  opposition,  find  by  inspection  two  simultaneous  longitudes, 
one  for  each  body,  that  differ  by  0°  or  180°.  The  corresponding  time 
of  the  first  will  be  that  of  conjunction,  and  of  the  second  of  opposition. 

§  650.  But  if  these  longitudes  are  not  to  be  found  in  the  tables,  take 
therefrom  two  consecutive  longitudes  for  each  body,  such,  that  those  of 
the  first  shall  differ  from  those  of  the  second,  in  order,  the  leavSt  possible. 
Then,  denoting  the  lesser  and  greater  longitudes  of  the  body  having  the 
greater  velocity  by  V  and  /'',  those  of  the  other  by  lt  and  ln  respectively, 
and  the  corresponding  times  by  t'  and  rf",  we  have,  because  the  longitudes 
of  each  are  given  for  the  same  epochs, 

(I"  _  /')  -  (/„  -  it)  :  (t"  -  t')  :  :  I,  -  V  :  *, 
whence 


m  which  x  denotes  the  interval  of  time  from  tf  to  conjunction.     And  de- 
noting the  ephemeris  time  of  conjunction  by  T<,,  we  have 


§  651.  Increasing  thr  longitudes  of  one  of  the  bodies  by  180°,  and  se- 


PROJECTION    OF    A    SOLAR    ECLIPSE. 


175 


lecting  those  of  the  other  to  differ  the  least  possible  from  these  increased 
longitudes,  then  will  Te  become  the  time  of  opposition. 

§  652.  Tc  is  the  local  time  on  the  meridian  for  which  the  ephemeris  is 
computed.  Denoting  the  longitude,  in  time,  of  any  other  meridian  west 
of  this  one  by  Z-,  and  the  local  time  of  conjunction  or  opposition  by  T. 
then  will 

T=TC-L (178) 


ANGLE  OF  POSITION. 

§  653.  The  angle  made  by  a  circle  of  latitude 
with  a  circle  of  declination  through  the  centre  of 
a  body,  is  called  the  angle  of  the  body's  position. 

To  find  this  angle,  let  P  be  the  pole  of  the 
ecliptic,  P/  that  of  the  equinoctial,  and  S  the  cen- 
tre of  the  body,  and  make 

X  =  90°—  P  S  =  latitude  of  the  body  ; 
S  =  90°-  P/  S  =  declination  of  the  body  ; 
a  =  P  P'  —  obliquity  of  the  ecliptic ; 

S  =  P  S  P'       =  angle  of  position  : 

then  will 
and 


Fig.  101. 


cos  rt  =  sin  X  .  sin  8  +  cos  X  .  cos  8 .  cos  S, 

„       cos  TX  —  sin  X  .  sin  8 
cos  S  = 


cos  X  .  cos  8 

654.  If  the  body  be  the  sun,  then  will  X  =  0,  and 

cos* 


cos  S  — 


cos  8 


(179) 


(180) 


PROJECTION  OF  A  SOLAR  ECLIPSE. 

§  655.  A  solar  eclipse  can  take  place  only  at  new  moon.  Find  the 
aphemeris  time  of  the  moon's  conjunction  with  the  sun.  Then,  by  the 
method  of  interpolation,  determine  the  sun's  true  longitude  and  hourly 
motion  in  longitude  ;  the  moon's  true  longitude  and  latitude,  and  hourly 
motion  in  longitude  and  latitude  ;  the  sun's  and  moon's  horizontal  paral- 
laxes, and  apparent  semi>diameters,  and  the  sun's  angle  of  position. 

§  656.  Conceive  a  cone  tangent  to  the  earth,  and  of  which  the  vertex 
is  at  the  sun.  A  section  of  this  cone,  by  a  plane  between  the  earth  and 
sun,  will  give  an  area  upon  which  the  sun's  centre  will  appear  tc  be  pro- 


176 


SPHERICAL    ASTRONOMY. 


jected  when  viewed  from  different  parts  of  the  earth.  A  section  at  the 
distance  of  the  moon  from  the  earth,  and  perpendicular  to  the  axis,  is 
called  the  circle  of  projection. 

§  657.  The  diurnal  rotation  of  the  earth  carries  an  observer  once  around 
his  parallel  of  latitude  in  24  hours ;  a  line  connecting  him  with  the  centre 
of  the  sun,  describes  an  entire  conical  surface  in  the  sam^  time,  and  a  sec- 
tion of  this  cone  by  the  circle  of  projection  will  be  the  paral  lactic  path  of 
the  sun  as  determined  by  the  axial  motion  of  the  earth.  This  ellipse  and 
the  relative  orbit  of  the  moon,  with  a  scale  of  time  on  each,  indicating  the 
simultaneous  positions  of  the  sun  and  moon,  being  constructed  upon  the 
plane  of  projection,  all  the  circumstances  of  a  local  solar  eclipse  may  easily 
be  predicted. 

Fig.  102. 


§  658.  Sun's  Parallactic  Path.— -Let  P  G  P'H  be  a  meridian  section 
of  the  earth  by  a  declination  circle  through  the  sun's  centre  at  S ;  E  the 
earth's  centre  ;  P  the  elevated  pole  ;  Gr  H  the  projection  of  the  equator ; 
B  A  that  of  the  observer's  parallel,  and  N N'  that  of  the  circle  of  projec- 
tion on  the  plane  of  the  section.  The  projection  A'B',  of  A  B  on  the 
circle  of  projection  by  the  lines  A  S  and  B  S,  will  be  the  conjugate,  and 
that  of  the  diameter  of  the  parallel,  which  is  perpendicular  to  A  B,  the 
transverse  axis  of  the  ellipse ;  the  first  being  in  and  the  second  perpendic- 
ular to  the  declination  circle  through  the  sun. 

§  659.  Make 

P  =  E N'U'=  moon's  horizontal  parallax  ; 
«  =  E  S  U'=  sun's 
/  =  A  E  H  =  reduced  latitude  of  place  ; 
d  —  ff£JS  =  sun's  declination  ; 
p  =  E  A       =  earth's  radius  ; 
u  =  number  of  seconds  in  radius. 


PROJECTION    OF    A    SOLAR   ECLIPSE. 

Draw  AC,  ED,  and  ^^perpendicular  to  E  S,  and  we 

A  C=  p  .sin  (I  -  d)  ;         B  D  =  p  .  sin  (I  +  d)  ; 
^  (7  =  p  .  cos  (I  -  d)  ;        E  D  =  p  .  cos  (I  +  d). 

Also,  Eq.  (28), 


177 

' 


whence 


From  the  figure, 


P  — * 
.__. 


S  C=  ES  —  EC=?  . p  .  cos  (/  —  o?). 

Then  in  the  triangles  S  C  A  and  S  MA', 

SC  :  SM  ::  AC  :  A'M, 
and  by  substitution 

rin(/-d)  ?.-* 


also, 


S  D  = 


1  _  I .  cos  (/  -  d) 


=  p  .  -  +  p  •  cos  (/  + 


and  in  the  same  way  as  above,  from  the  triangles  S  D  B  and  S  M  B', 

sin  (l  +  d)  P-« 


But  ie  can  never  exceed  9",  and  u  is  equal  to  206264".8,  so  that  the 
terms  into  which  *r  -7-  w  enters  as  a  factor  may  be  neglected,  and  we  have 

n(l-d).~- (181) 

JT 

Qn  +  d).^-  ....'.    (182) 


MAf  = 


From  which  we  see  that  the  length  of  the  projection  of  any  dimension 
at  the  earth,  and  parallel  to  the  circle  of  projection,  is  found  by  multiply- 
ing this  dimension  by  (P  —  t)  -r-  P. 

§  660.  Denoting  the  conjugate  axis  A'B'  by  2  6,  we  have 


=  M&-  MA', 
12 


SPHERICAL   ASTRONOMY. 


and  by  substitution, 


Also, 


b  =  p  .  cos  / .  sin  d  . 


P  - 


(183) 


FA  =  p  .  cos  I ; 

and  because  that  diameter  of  the  parallel  of  latitude,  which  is  perpendic- 
ular to  A  B,  is  parallel  to  the  plane  of  projection,  we  have,  denoting  the 
semi-transverse  axis  of  the  ellipse  by  a, 

P  —  « 


a  =  p  .  cos  I . 


(184) 


And  denoting  the  distance  M  Fr  from  the  centre  of  the  circle  of  projec- 
tion to  that  of  the  ellipse  by  JT,  we  have,  taking  half  sum  of  equations 
(181)  and  (182) 


Y  =  p > .  sin  I .  cos  d  . 


(185) 


§  661.  Revolve  the  parallel  of  latitude  about  AE  till  it  coincides  with 
the  meridian  section.  When  the  observer  is  at  A,  it  is  to  him  apparent 
noon ;  when  at  B,  apparent  midnight;  when  at  0,  the  angle  OF  A  is  the 
apparent  hour  angle  of  the  sun,  and  therefore  local  apparent  time. 

Fig.  102  bis. 


Draw  0  T  perpendicular  to  A  B,  and  S  L  through  the  point  T.      The 
projection  of  F  L  will  give  the  distance  of  the  sun  from  the  transverse, 


and  that  of  0  This  distance  from  the  conjugate  axis  of  his  elliptical  path. 


Denote  the  first  by  y,  the  second  by  ar,  and  the  hour  angle  0  F  A  by  &. 
-jThen 

FO  =  FA  =  ?.coal-, 

0  T  =  p  .  cos  I  .  sin  h  ; 
FT  =  p.  cos  /.cos  h\ 


PROJECTION   OF   A   SOLAR   ECLIPSE. 


179 


and  since  F L  T  is  sensibly  a  right  angle,  the  value  of  E  S  U,  which  is 
much  greater  than  E  S  T,  never  exceeding  9" ;  and  because  FTL  = 
UEP  =  d,  we  have 

F  L  =  F  T.  sin  d  —  p  .  cos  I .  cos  h  .  sin  d  ; 

and  projecting  F  L  and  0  T  on  the  circl  3  of  projection,  there  will  result 


.    .    ,       ,  p-«r 

y  =  p  .  cos  /  .  sin  d  .  cos  h  .  — - — 


x  =  p  .  cos  I .  sin  h  . 


P- 


(186) 

(187) 


But  p  -r-  P  is  the  linear  subtense  of  the  unit  of  arc  in  which  P  is  ex- 
pressed— say  one  minute.  Calling  this  distance  unity,  equations  (183), 
(184),  (185),  (186),  and  (187)  may  be  written 

b  =  cos  I.  sin  d  (P'  —  «') (188) 

a=<x»l.(P'-«')  . (189) 

Y=sml.co&d.(P'  —  if') (190) 

y  =  cos  / .  sin  d .  cos  h  .  (P/  —  if')      ,     .     .  (191) 

x  =  cos  I .  sin  h  .  (P'  —  **) (192) 

§  662.  Let  G  be  the  centre  of  the  circle  of  projection,  CIV  the  trace 
ui  a  circle  of  declination  through  the  sun's  centre  on  the  plane  of  projec- 
tion. Take  the  distance  Coequal  to  F,  Eq.  (190);  through  /^  draw 
A  A  perpendicular  to  C  N,  and  make  FA  —  FA  —  a,  Eq.  (189) ;  take 
F  E  =  F  B'  —  6,  Eq.  (188) ;  and  making,  successively,  h  equal  to  15°, 
30°,  45°,  <fec.,  in  Eqs.  (191)  and  (192),  construct  the  corresponding  hour 


180 


SPHERICAL    ASTRONOMY. 


Fig.  103  bia. 


points  of  the  parallactic  path  of  the  sun's  centre.  The  geometrical  con- 
struction of  this  path  is  indicated  in  the  figure. 

§  663.  Moon's  geocentric  relative  orbit. — Substitute  in  Eq.  (180)  the 
values  of  itf  and  £,  and  make  the  angle  NC  L  equal  to  the  resulting  value 
of  S ;  the  line  C L  will  be  the  trace  of  a  circle  of  latitude  on  the  circle 
of  projection.  Make  CD  equal  to  the  moon's  latitude  at  conjunction, 
and  draw  D  E  perpendicular  to  C  L  and  equal  to  the  excess  of  the  moon's 
hourly  motion  in  longitude  over  that  of  the  sun  ;  draw  #  J7  perpendicular 
to  E  D  and  make  it  equal  to  the  moon's  hourly  motion  in  latitude  ; 
through  H  and  D  draw  an  indefinite  straight  line  ;  this  line  will  represent 
the  moon's  geocentric  relative  orbit  on  the  plane  of  the  circle  of  projection. 

§  664.  Scale  of  time  on  the  Moon's  geocentric  relative  orbit. — Make 

m  =  moon's  hourly  motion  in  longitude  ; 
•      n  =       "         "  "  latitude ; 

*  =  sun's  hourly  motion  in  longitude  ; 
i  =  L  D  H—  inclination  of  the  geocentric  relative  orbit  to  circle  of 

latitude  through  the  moon  at  conjunction. 
m'  =  moon's  hourly  motion  on  relative  orbit : 


then 


cot  i  = 


in  — 


m!  =  (m  —  s) .  cosec  i 


(193) 
(194^ 


From  the  ephemeris  time  of  conjunction  take  the  longitude  of  the  place 
in  time,  the  remainder  will  be  the  local  time  at  which  the  moon's  centre 


PROJECTION  OF  A  SOLAR  ECLIPSE. 


181 


is  at  D.  Let  e  denote  its  excess  above  the  next  preceding  whole  hour,  and 
a  the  distance  from  D  to  the  moon's  centre  at  that  hour ;  then  will 

a  =  (m  —  s)  .  cosec  i  .  e (195) 

and  laying  off  this  distance  from  D  to  the  west  on  the  relative  orbit,  we 
have  one  point,  and  the  entire  hour  on  the  scale  of  time  indicating  the 
positions  of  the  moon  at  different  times.  From  this  point  lay  off  distances 
to  the  west  and  east  equal  to  the  value  of  m'  in  Eq.  (194),  and  there  will 
result  a  series  of  points  corresponding  to  the  entire  hours,  as  indicated  in  the 
figure.  In  the  example  before  us,  the  local  time  of  conjunction  is  about 
2h.33  P.  M.,  the  hour  point  2  falling  about  0.33  of  m'  to  the  west  of  D. 

§  665.  Parallactic  relative  orbit  of  the  Moon. — The  apparent  path  of 
the  moon  in  reference  to  the  sun's  centre,  as  seen  from  the  earth's  surface, 
is  the  moon's  relative  parallactic  orbit.  To  construct  this  path,  draw 
through  the  sun's  places  on  his  parallaetic  path  and  the  moon's  places  on 
her  geocentric  relative  orbit,  at  the  same  hours,  lines  respectively  parallel 
and  perpendicular  to  ON,  Fig.  103  ;  a  series  of  rectangles  will  thus  be 
formed  :  the  sides  of  these  rectangles  which  terminate  in  the  sun's  places 
will  be  the  co-ordinates  of  the  moon's  parallactic  relative  orbit,  in  refer- 
ence to  the  sun's  centre,  regarded  as  fixed. 

Fig.  104 


For  example,  draw  S  Y  and  MX  parallel  and  S  X  and  M  Y  perpen- 
dicular to  G N\  then  making  S  X  and  S  Y,  in  Fig.  104,  respectively 
equal  and  parallel  to  S  Jf  and  S  Ym  Fig.  103,  and  drawing  X M  and 
YM  respectively  parallel  and  perpendicular  to  S  N,  their  intersection  M 
will  give  a  point  in  the  parallactic  relative  orbit  of  the  rnoon.  This  point 
in  the  example  of  the  figure  corresponds  to  3h.  Other  points  being  con- 
structed in  the  same  way,  the  orbit  sought  and  its  scale  of  time  may  be 
completed. 

§  666.  Beginning,  Ending,  and  Greatest  Obscuration. — The  hour  in- 
tervals on  the  scale  of  time  being  suitably  subdivided,  with  S  as  a  centre 


182 


SPHERICAL   ASTRONOMY. 


and  radius  equal  to  the  sum  of  the  apparent  semi-diameters  of  the  sun  and 
moon,  describe  an  arc  cutting  the  parallactic  relative  orbit  of  the  moon  in 
the  points  m  and  w», ;  the  corresponding  numbers  on  the  scale  will  be  tho 
times  of  beginning  and  ending,  the  former  being  at  m  and  the  latter  at  w»t. 
From  S  draw  Sm2  perpendicular  to  the  nearest  portion  of  relative  orbit, 
the  number  at  ma  on  the  scale  will  give  the  time  of  greatest  obscuration. 
Also  Q  Q'  will  be  the  quantity  of  the  eclipse. 

§  667.  The  Angle  of  First  Contact. — With  S  as  centre  and  radius 
equal  to  the  sun's  apparent  semi-diameter,  describe  the  circumference 
VWZ;  with  m  and  mt  as  centres  and  radii  equal  to  the  moon's  apparent 
semi-diameter,  describe  two  other  circumferences.  The  tangential  points 
E"  and  E"  will  be  those  of  first  and  last  contact  respectively.  The  for- 
mer of  these  it  is  important  to  know  in  advance  to  guide  the  observer's 
attention  in  his  efforts  to  note  the  actual  time  at  which  the  solar  eclipse 
begins.  The  angular  distance  of  this  point  from  the  highest  point  or  ver 
tex  of  the  sun,  measured  around  the  solar  disk,  is  called  the  angle  of  firal 
tontact. 

§  668.  To  find  this  angle,  transfer  the  point  m,  which  is  at  2k.5,  to  the 
corresponding  point  S'  on  the  parallactic  path  of  the  sun,  Fig.  103,  and 
draw  C  Sf.  This  line  will  represent  the  trace  of  a  vertical  circle  through 
the  sun's  centre  on  the  circle  of  projection,  because  every  such  circle  must 
pass  through  the  earth's  centre,  and  therefore  contain  the  point  C.  Ma- 
king the  angle  K'SV'm  Fig.  104,  equal  to  1C  OS' in  Fig.  103,  the  point 
V  will  be  the  vertex  of  the  solar  disk  and  VSm  the  angle  of  first  contact. 

§  669.  By  this  construction,  the  beginning  of  a  solar  eclipse  may  be 
predicted  within  one  minute  of  its  actual  occurrence,  and  this  will  in  gen 
eral  be  sufficient  for  the  practical  purpose  of  indicating  the  time  to  look  for 
the  instant  of  first  contact — one  of  the  most  important  elements,  as  we  shall 
presently  see.  in  the  determination  of  terrestrial  longitude. 


PRuJEJTIOX    OF   A   LUNAR   ECLIPSE. 


183 


For  a  full  and  complete  investigation  of  the  whole  subject  of  eclipses, 
occupations,  and  transits,  see  Appendix  XL,  which  consists  entirely  of  the 
admirable  paper  of  Mr.  Woolhouse,  first  published  as  an  appendix  to  the 
British  Nautical  Almanac  for  1836. 


PROJECTION  OF  A  LUNAR  ECLIPSE. 

§  670.  During  a  lunar  eclipse,  the  earth's  shadow  rests,  as  it  were,  upon 
the  actual  surface  of  the  moon,  and  deprives  it  of  a  portion  of  the  solar  light 
which  it  would  otherwise  receive  and  reflect  to  a  spectator  on  the  earth. 

§  671.  A  section  of  the  earth's  shadow  at  the  moon  will  have  the  same 
parallax  as  the  moon,  both  being  at  the  same  distance  from  the  earth ;  and 
regarding  ie  as  appertaining  to  the  centre  of  this  section,  P  —  it  will  be 
zero;  Y,  a,  6,  #,  and  y  will,  equations  (188)  to  (192),  be  zero,  and  the 
parallactic  path  of  the  earth's  shadow  on  the  plane  of  projection  will  re- 
duce to  a  point. 

§  672.  Regarding,  therefore,  S,  in  Fig.  104,  as  the  centre  of  a  section 
of  the  earth's  shadow  at  the  moon,  and  V  W  Z  as  its  circumference,  then 
will  m  m2  m-,  represent,  not  the  parallactic,  but  the  geocentric  relative  orbit 
of  the  moon.  . 

§  673.  Hence,  to  project  a  lunar  eclipse,  find  from  the  ephemeris  the 
time  of  opposition  or  full  moon,  the  corresponding  values  of  the  moon's 
latitude,  hourly  motion  in  latitude,  longitude,  hourly  motion  in  longitude, 
horizontal  parallax,  and  apparent  semi-diameter ;  also  the  sun's  longitude, 
hourly  motion  in  longitude,  horizontal  parallax,  and  apparent  seini- 
diameter. 

Fig.  106. 


§  674.  Then,  with  S  as  centre  and  radius  equal  to  that  of  tho  earth's 
shadow  at  the  moon,  Eq.  (148),  describe  the  circumference  VWZ.     Draw 


184 


SPHERICAL    ASTRONOMY. 

Fig.  105  bis. 


S  JV  to  represent  an  arc  of  a  circle  of  latitude ;  make  S  0  equal  to  tne 
moon's  latitude  at  opposition,  and  construct  the  moon's  relative  geocentric 
path  and  scale  of  time,  as  in  §  663-4.  With  S  as  centre  and  radius 
equal  to  that  of  the  earth's  shadow,  increased  by  the  moon's  apparent 
semi-diameter,  describe  an  arc  cutting  the  relative  orbit  in  B  and  E,  and 
let  fall  from  S  the  perpendicular  S  M  on  the  relative  orbit.  The  numbers 
on  the  scale*  of  time  at  B,  M,  and  E  will  give  respectively  the  time  of  be- 
ginning, middle,  and  ending  of  a  lunar  eclipse. 

TIME  OF  DAY. 

§  675.  The  imperfection  incident  to  all  machinery  makes  it  impossible 
to  construct  a  time-piece  to  run  accurately  to  mean  solar  or  sidereal  time. 
The  best  efforts  result  only  in  approximations ;  and  when  these  are  made 
to  the  utmost  attainable  limits,  it  must  remain  for  astronomical  observa- 
tions and  computations  to  do  the  rest  by  detecting  and  applying  from  time 
to  time  the  amount  of  error. 

§  676.  Time  by  Meridian  Transits. — When  the  sun's  centre  is  on  the 
meridian  it  is  apparent  noon ;  and  twelve  hours  or  twenty-four  hours  (ac- 
cording as  it  is  civil  or  astronomical  time),  corrected  for  the  equation  of 
time,  gives  the  mean  solar  time  at  the  same  instant.  When  a  star  is  on 
the  meridian,  its  right  ascension,  in  time,  is  the  sidereal  time  at  that  in- 
stant. And  the  most  simple  and  accurate  method  of  finding  the  time, 
and,  therefore,  the  error  of  a  time-piece,  is  to  note  the  indications  of  the 
latter  when  the  west  and  east  limbs  of  the  sun  or  a  star  cross  the  wires  of 
a  transit  instrument  properly  adjusted  to  the  meridian.  A  mean  of  these 
indications  gives  the  watch  time  of  the  transit  of  the  sun's  centre  in  the 
first  3ase,  or  that  of  the  star  in  the  second.  The  difference  between  the 
first  and  the  mean  solar  time  gives  the  error  on  mean  solar  time ;  and 


TIME    OF   DAY. 


185 


Fig.  106. 


between  the  second  and  the  star's  right  ascension,  the  error  on  sidereal 
time. 

It  is  not,  however,  always  possible  to  use 
the  transit,  and  recourse  must  be  had  to  ob- 
servations off  the  meridian. 

§  677.  Solar  Time  by  a  single  Altitude. 
— To  find  the  mean  solar  time  and  error  of 
time-keeper  from  one  observed  altitude  of 
the*  sun :  Let  P  be  the  pole,  Z  the  zenith, 
and  S  the  place  of  the  sun's  centre ;  and 
make 

a  =  true  altitude  of  sun's  centre  =  90°  —  Z  S ; 
d  —  true  declination  of  sun     .     =  90°  —  P  S ; 
I  =  latitude  of  the  place    .     .     =  90°  —  P  Z ; 
P  =  hour  angle  Z  P  S ; 
s  =  error  of  the  watch ; 
Tw  =  watch  time  of  observation ; 
Ta  =  apparent  time  of  observation ; 
Tm  =  mean  time  of  observation ; 
E  =  equation  of  time. 

Then,  in  the  triangle  P  Z  S,  we  have  from  spherical  trigonometiy, 


sin 


=  ± 


in  which 


whence 


cos  I  .  cos 


=  270°  -  (a  +  d  +  I) 
2 


(196) 


p  = 


cos    .  cos 


and 


(198) 
(199) 


The  latitude  of  the  place  of  observation  is  supposed  known.  The  value 
of  s  requires,  in  addition,  the  values  of  a,  e£,  E,  and  Tw  to  be  known. 

To  find  a  and  Tv,  measure  with  a  sextant  and  artificial  horizon,  or  other 
instrument  for  taking  altitudes,  the  altitude  of  the  upper  or  lower  limb  of 
the  sun — say  the  lower — and  note  the  precise  indication  of  the  watch  at 
the  instant.  Tw  is  thus  found,  an  1 


186  SPHERICAL   ASTRONOMY. 

a  =.  observed  altitude  of  the  lower  limb  —  refraction  -f  apparent  semi- 
diameter  of  the  sun  -}-  the  sun's  parallax  in  altitude. 

To  find  d,  convert  the  watch  time — supposed  not  greatly  in  error,  other- 
wise the  estimated  local  time  of  observation — into  the  corresponding  local 
time  on  the  meridian  for  which  the  ephemeris  of  the  sun  is  computed,  say 
that  of  Greenwich  ;  take  from  the  ephemeris  the  sun's  declination  for  the 
next  preceding  mean  noon,  and  also  the  hourly  change  in  declination; 
then 

d  =  declination  at  the  preceding  noon  ±  its  hourly  change  multiplied 
into  the  Greenwich  time  of  observation  ; 

the  upper  sign  to  be  used  when  the  declination  is  increasing,  and  the  lower 
when  decreasing. 

To  find  E,  take  from  the  ephemeris  the  equation  of  time  for  the  next 
preceding  mean  noon,  and  the  hourly  change ;  then 

E  =  equation  at  the  preceding  mean  noon  dr  its  hourly  change  into  the 
Greenwich  time  of  observation. 

§  678.  It  is  usual  to  avoid  the  correction  for  semi-diameter  by  clamp- 
Ing  the  instrument  at  some  assumed  altitude,  and  noting  the  time,  by  the 
watch,  that  the  upper  and  lower  limb  of  the  sun  attain  this  altitude. 
The  mean  of  these  times  will  be  the  time  when  the  sun's  centre  had  thia 
same  altitude,  and  it  will  only  be  necessary  to  correct  the  observed  altitude 
for  refraction  and  parallax. 

§  679.  It  is  to  be  remembered  that  P  has,  Eq.  (197),  the  double  sign  : 
the  positive  answers  to  the  case  in  which  the  hour  angle  is  west,  or  the 
observation  is  made  in  the  afternoon ;  and  the  negative  to  that  in  which 
the  hour  angle  is  east,  or  the  observation  is  made  in  the  morning.  In  this 
latter  case,  -j^  P  must  be  replaced  by  12h  —  T\  P  if  civil,  or  24h  —  T^  p 
if  astronomical  time  be  sought. 

This  process  for  finding  the  error  of  a  time-piece  is  called  the  method  of 
nngle  altitudes. 

§  680.  Sidereal  time  by  a  single  Altitude  of  a  Star. — Proceed  exactly 
as  in  §  677-9,  using  the  declination  and  observed  altitude  of  the  star  for 
those  of  the  sun,  correcting  the  altitude  for  refraction  only,  and  find  the 
value  of  -jJj  P,  to  which  add  the  right  ascension  of  the  star  as  found  from 
the  catalogue ;  the  sum  will  be  the  sidereal  time. 

§  681.  The  rules  for  converting  solar  into  sidereal  tame,  and  the  re- 
verse, together  with  tables  for  facilitating  the  same,  are  given  in  the  solai* 
ephemeris. 


TIME    OF   DAY. 

§  682.  Time  of  Sunrise  and  Sunset.  —  At  the  instant  of  apparent  sun- 
rise, the  sun's  centre  is  in  the  horizon  ;  the  apparent  altitude  of  its  lower 
limb  is  equal  to  minus  its  apparent  semi  -diameter,  and  a,  in  Eq.  (197), 
becomes  the  difference  between  the  horizontal  refraction  and  parallax. 
Making  this  substitution  in  Eq.  (197)  we  find  P,  and  this  in  Eq.  (198) 
gives  Tm. 

"§  683.  Time  by  Equal  Altitudes.  —  If  the  sun  retained  unchanged  his 
declination,  equal  altitudes  would  correspond  to  equal  hour  angles,  and  the 
half  sum  of  the  watch  times,  augmented  by  6h  when  the  dial-plate  is  di- 
vided into  12,  and  12h  when  divided  into  24  hours,  would  give  the  watch 
time  of  apparent  noon.  Twelve  or  twenty  -four  hours,  depending  upon  the 
dial-plate,  corrected  for  the  equation  of  time  would  give  the  mean  time  of 
apparent  noon,  and  the  difference  between  this  and  the  corresponding 
watch  time  would  give  the  error. 

But  the  sun  is  ever  changing  his  declination,  and  when  the  effect  of  the 
change  is  to  lessen  his  distance  from  the  elevated  pole  between  the  obser- 
vations, his  hour  angle  in  the  morning  will  be  less  than  in  the  afternoon  at 
equal  altitudes  ;  the  watch  time  of  apparent  noon,  as  above  found,  would 
be  too  late,  and  must  be  corrected  by  subtracting  therefrom  half  the  excess 
of  the  evening  over  the  morning  hour  angle.  Conversely,  when  the  effect 
is  to  'augment  the  distance  from  the  elevated  pole,  the  morning  hour  angle 
will  exceed  the  evening  ;  the  watch  time  of  apparent  noon,  as  found  by  the 
rule,  will  be  too  early,  and  must  be  augmented  by  half  the  excess  of  the 
morning  over  the  evening  hour  angle. 

Denoting  this  correction  by  t{,  its  value  in  seconds  of  time  will,  Appen- 
dix XII.,  be  given  by 


or  making 


1440  sin  7  %t 

t 
1440  tan  7  %t  ~ 

t,  =  +  A  .  S  .  tan  I  -  B  .  6  .  tan  d  .    .     .    .    (200) 
In  which 

t  =  the  interval  of  time  between  the  observations  in  hours; 

/  =  latitude  of  place ; 

d  =  declination  of  sun  at  noon  of  the  day  , 


188  SPHERICAL   ASTRONOMY. 

§  =  double  daily  variation  in  declination,  or  change  from  noon  of 
preceding  to  noon  of  following  day. 

The  value  of  t  will  be  subtractive  for  apparent  noon,  and  additive  for 
apparent  midnight,  when  d  is  positive,  and  conversely. 

The  logarithms  of  the  values  of  A  and  B  are  given  in  Table  IV.,  for 
every  two  minutes,  from  two  hours  up  to  twenty-three ;  the  latitude  of  the 
place  must  be  known;  the  declination  is  found  as  in  §  677,  and  6  is  ob- 
tained from  the  ephemeris. 

§  684.  Change  of  atmospheric  temperature  and  of  pressure. — In  what 
precedes  it  is  supposed  that  when  the  measured  altitudes  are  equal,  the 
true  altitudes  are  so  likewise ;  but  this  depends  upon  the  state  of  the  air 
remaining  the  same  between  the  observations.  If  the  barometer  and  ther- 
mometer vary,  the  refraction  will  vary,  and  the  true  altitudes  will  be  un- 
equal when  the  observed  are  equal.  A  further  correction  becomes  there- 
fore necessary,  and  its  value  is  the  increment  or  decrement  of  the  hour 
angle  of  the  sun,  which  would  change  his  altitude  by  a  quantity  equal  to 
the  difference  between  the  morning  and  evening  refraction.  The  amount 
of  this  correction,  denoted  by  tlt  and  expressed  in  seconds  of  time,  is, 
Appendix  XIII.,  given  by 

'"  =  &  '  cosl7w*  d™n  P (2°1} 

in  which 

r  =  morning  refraction,  in  seconds  of  arc  ; 
r'  =  evening  refraction,  in  seconds  of  arc  ; 
P  =  half  the  interval  between  the  observations  in  arc  ; 
a  =  altitude  of  sun  ; 
d  =  declination ; 
I  =  latitude  of  place. 

This  process  for  finding  the  time  of  day,  or  error  of  a  time-piece,  is  callea 
the  method  of  equal  altitudes  ;  and  the  value  of  t,,  in  Eq.  (200),  is  called 
the  equation  of  equal  altitudes. 

§  685.  The  altitudes  should  be  taken  on  or  near  the  prime  vertical, 
since  in  that  position  the  altitudes  change  most  rapidly. 

AZIMUTHS. 

§  686.  In  surveys  and  geodetic  operations,  it  is  necessary  to  determine 
the  bearings  of  objects  in  reference  to  the  meridian  of  the  station  from 
which  they  are  seen.  These  bearings  are  measured  by  the  angles  at  the 


AZIMUTHS. 


189 


Fig.  10T. 


zenith  included  between  the  vertical  circles  through  the  objects  and  the 
meridian. 

§  687.  True  Bearing.  —  To  find  the  true  bearing  of  an  object  from  a 
given  station.  Take  the  instrumental  bearing  of  the  object  and  of  some 
heavenly  body,  and  add  their  difference  to  the  true  azimuth  of  the  body  ; 
the  sum  will  be  the  true  azimuth  of  the  object. 

To  find  the  true  azimuth  of  a  heavenly  body,  note  the  time  its  instru- 
mental bearing  is  taken. 

Let  Z  be  the  zenith,  P  the  pole,  and  S 
the  heavenly  body,  say  the  sun  or  a  star  ; 
make 

P  =  hour  angle  Z  P  S  ; 
I  =  latitude  of  place  ; 
d  =  declination  of  body  ; 
Tm  =  mean  solar  time  of  observation  ; 
E  =  corresponding  equation  of  time  ; 
Ta  =  apparent  solar  time  of  observa- 

tion ; 

T^  =  sidereal  time  of  observation  ; 
AQ  =  right  ascension  of  mean  sun  ; 
AQ  =  right  ascension  of  a  star. 

Then  with  the  sun  and  mean  solar  time, 
we  have 

P  =  15  .  Ta  =  15  (Tm  ±  E)     (202) 
With  the  mean  solar  time  and  star, 

P  =  15  (Tm  +  AQ  -  A%)        (203) 
With  the  sidereal  time  and  the  sun, 

P  =  15  (T+—  AQ±JB)  .  .   (204) 
With  the  sidereal  time  and  »tar, 

P=15(r.-J.)....    (205) 
Also  make 


=  P  S  =  90°  -  d  ; 

A  =  the  angle  P  Z  S  =s  180°  —  azimuth  of  the  body  ; 
i=    "       «      ZSP. 


190  SPHERICAL  ASTRONOMY. 

Then  in  the  triangle  Z  P  S,  from  Napier's  Analogies, 


^    ^(A-^cotJP.  .     .     .     (207) 

Irom  which  A  becomes  known.     The  angle  f  ,  or  that  at  the  body  is  tech- 
nically called  the  angle  of  variation^ 

§  688.  The  north  star,  called  Polaris,  is  often  advantageously  used  for 
this  purpose,  particularly  when  it  has  its  greatest  eastern  or  western  elon- 
gation. At  that  time  the  vertical  circle  through  the 
star  is  tangent  to  its  diurnal  path,  and  its  diurnal 
motion  will  be  in  altitude  alone,  and  not  at  all  in  azi- 
muth, thus  affording  time  for  taking  a  series  of  azimu- 
thal  distances. 

To  find  the  time  when  Polaris  or  other  circumpolar 
star  has  its  greatest  elongation,  observe  that  the  angle 
of  variation  is  at  that  instant  90°,  and  in  the  right-an- 
gled triangle  P  Z  S,  right-angled  at  S,  we  have 

cos  P 
and 

TX  =  A%  +  jSg.  .  cos 
Also 

. 

sin  A  := 

§  689.  When  the  bearing  of  the  sun  is  taken,  the  line  of  collimation 
must  be  directed  to  one  or  the  other  extremity  of  his  horizontal  diameter  ; 
to  the  bearing  of  which  must  be  added  the  horizontal  semi-diameter  re- 
duced to  the  horizon,  which  is  equal  to  the  tabular  semi-diameter  divided 
by  the  cosine  of  the  sun's  altitude. 

§  690.  Variation  of  the  Compass.  —  From  the  foregoing  it  will  be  easy 
to  find  the  variation  of  the  compass,  or,  as  it  is  frequently  called,  the  dec- 
lination of  the  magnetic  needle.  For  this  purpose  it  will  be  sufficient  to 
take  the  magnetic  bearing  of  some  heavenly  body  and  note  the  time. 
Then  from  the  time  and  equations  (206)  and  (207),  computing  the  true 
azimuth,  and  taking  the  difference  between  the  magnetic  and  true  azi- 
muths, the  problem  is  solved. 

Or,  if  the  true  bearing  of  any  terrestrial  object  be  known,  we  have  only 
to  subtract  it  from  the  magnetic  bearing,  as  determined  by  the  compass, 
to  obtain  the  same  result, 


S'1  [tan  5 

.  cot  9]  . 

V^wu; 

.     .     .     (209) 

sin  d 

(210} 

sm  9 

MERIDIAN    PASSAGE 


191 


§  691.  At  sea,  or  on  land  where  the  ob- 
server is  surrounded  by  prairies  or  extended 
plains,  it  is  usual  to  take  the  magnetic 
bearing  of  the  sun's  centre,  by  observing 
alternately  the  opposite  horizontal  limbs  at 
the  time  of  rising.  Then  in  the  triangle 
Z  S  P,  we  have 

Z  S  =  90°  +  refraction  -  parallax  =  £  ; 
and  making 


Fig.  110. 


\ 


cos  4  A  = 


sin 


sin  9 


It  will  be  sufficient  to  regard  the  horizontal  refraction  and  parallax  as 
constant,  and  the  former  equal  to  33'  45"  and  the  latter  to  8";  thus  ma- 
king 

£=  90°  33'  37". 


MERIDIAN  PASSAGE. 

§  692.  It  is  often  desirable  to  know  in  advance  what  will  be  the  indi- 
cation of  a  sidereal  or  mean  solar  time-piece  at  the  instant  a  given  body  is 
on  the  meridian.  This  indication  will  measure  the  hour  angle  of  the  ver- 
nal equinox,  or  of  the  mean  sun  at  the  instant,  according  as  the  time- 
keeper is  running  to  sidereal  or  mean  solar  time. 

§  693.  Time  of  Meridian  Passage.  —  To  find  the  local  mean  solar  time 
of  a  given  body  coming  to  the  meridian,  make 

t  =  the  time  required  ; 

A1  '„  =  right  ascension  of  mean  sun  at  this  time  ; 
A'  =  right  ascension  of  the  body  at  the  same  instant  ; 
As  =  right  ascension  of  mean  sun  at  Greenwich,  mean  noon  next  pre- 

vious ; 

A  =  right  ascension  of  the  body  at  the  same  instant  ; 
s  =  hourly  change  of  mean  sun  in  right  ascension  ; 
m  =  hourly  change  of  body  in  right  ascension  ; 

I  =  longitude  of  place  in  time. 
Then  the  time  at  Greenwich  corresponding  to  the  'ocal  time  £,  will  be 

*  +  /  ;  and 

A',  =  A.  +  s  (t  +  /), 
A'  =  A   +  m  (t  +  /)  ; 


192  SPHERICAL    ASTRONOMY. 

and  since  all  the  elements  are  expressed  in  time,  the  difference  of  right 
ascension  of  the  sun  and  body,  when  the  latter  is  on  the  meridian,  must 
equal  t;  whence 

A' -  A',  =  A  +  m  (t  +  /)  -  A.  —  s  (t  +  I)  =  t; 
or 

t=A-A.  +  l(m~s)  (212) 

1   __    (m  _  8) 

in  which  A,  As,  m,  and  s  must  be  expressed  in  the  same  unit,  say  hours. 
§  694.  If  the  body  should  be  a  star,  then  will  m  =  0,  and 

t  =  ±^±=ll     ......     (213) 

695.  If  a  planet  with  retrograde  motion,  m  would  change  its  sign,  and 

t  =  A~^~'^  +  t) (2H!) 

1  +  m  -h  s 

§  696.  If  the  sidereal  time  were  asked  for,  then  would  A,  =  0, 
»  =  0,  and 

(  =  £±_^ (215) 

1  —  m 

and  if  the  body  be  a  star,  then  m  =  0,  and 

t  =  A. 

REDUCTION  TO  THE  MERIDIAN. 

§  697.  Some  of  the  most  important  astronomical  determinations  de 
pend  upon  the  measured  zenith  distances  or  altitudes  of  a  body  when  on 
the  meridian ;  but  these  measurements  it  is  not  always  convenient  nor 
possible  to  make,  and  besides  it  is  desirable  to  multiply  measurements 
as  much  as  possible  to  secure  the  advantages  of  a  general  average  in  elim- 
inating errors  of  observations.  The  purpose  of  the  next  proposition  is, 
therefore,  to  pass  from  a  measured  zenith  distance  or  altitude  taken  when 
the  body  is  off  the  meridian  to  what  it  would  have  been  had  the  body 
been  on  that  circle. 

The  difference  between  any  two  zenith  distances,  applied  with  the  proper 
sign  to  either,  will  give  the  other ;  and  when  one  is  the  meridian  zenith 
distance,  this  difference  is  called  the  reduction  to  the  meridian. 

§  698.  Reduction  to  the  Meridian. — To  find  the  reduction  to  the  rae 
ridian. 


REDUCTION    TO   THE    MERIDIAN. 

Fig.  ill. 


193 


Let  P  be  the  pole,  Z  the  zenith,  S  a 
star,  S  M  an  arc  of  the  star's  diurnal  circle 
cutting  the  meridian  in  M,  S  0  the  arc  of 
a  horizontal  circle  through  the  star,  and 
cutting  the  meridian  in  0.  Make 

x  =  Z  S-ZM  =  Z  0  -  Z  M=  re- 

duction to  meridian  ; 
I  =  latitude  of  place  ; 
d  =  declination  of  star  ; 
P  =  hour  angle  Z  P  S  ; 
z  =  zenith  distance  Z  S. 

Then  because 

P  Z  =  90°  -  I  ;  P  S  =  90°  -  d  ; 

we  have  in  the  triangle  P  Z  S 

cos  z  =  sin  I  .  sin  d  +  cos  I  .  cos  G?  .  cos  P  ; 
but 

cos  P  =  1  —  2  .  sin2  i  P  ; 

and  substituting  this  we  get 

cos  z  =  sin  I  .  sin  d  +  cos  I  .  cos  d  —  2  cos  /  .  cos  d  sin2  ^  P, 
=  cos  (I  —  d)  —  2  cos  Z  .  cos  d  .  sin2  ^  P. 

But  Z  M  =  I  —  d  ;  and  2  =  x  +  Z  —  d  ;  and  therefore, 
cos  z  =  cos  #  .  cos  (/  —  -  •  rf)  —  sin  x  sin  (J  —  d)  ; 
cos  #  —  1  —  ^  ar2  +  ,  <fec.  ; 

and  if  the  observations  be  made  near  the  meridian,  x  will  be  very  small, 
and  its  powers  higher  than  the  second  may  be  neglected.  Making  this 
supposition,  writing  the  arc  for  its  sine,  and  substituting  the  value  of 
cos  x  above,  we  have 

cos  z  —  (1  —  \  a*)  .  cos  (I  —  d)  —  x  .  sin  (/  —  d). 

Equating  these  values  of  cos  z?  there  will  result 

J  «•  .  cos  (l  —  d)  +  x  sin  (I  —  d)  =  2  cos  I  .  cos  d  .  sin2  £  P  .  .  (216) 

In  consequence  of  the  small  value  of  a?,  it  will  be  sufficient  for  all  prac- 
tical purposes  to  make  an  approximate  solution  of  this  equation  ;  for  thia 
purpose  write  it 

2  cos  I  .  cos  d 


also, 


x  = 


sin  (I  -  d) 


.sin*  J  P-cotan(/-o?)  .  J 


(217) 


13 


(218) 
v       ' 


194  SPHERICAL   ASTRONOMY. 

neglecting  the  term  involving  the  second  power  of  #, 

2  cos  I  .  cos  d      .  2 

1  sin(Z-d)    'S11  ¥ 

and  this  in  Eq.  (217)  gives 

cos  / .  cos  d   r  _.     /cos  / .  cos  dV 

x  =  -r-n -7T  •  2  sin2!  P  _  cot  (Z-  d)  .  (  -— - — )  .  2  sm4  i  P, 

sm  (I  —  o?)  \sm  (I  —  d)  / 

and  making,  in  order  to  find  x  in  seconds  of  arc, 

2  sin2  IP  _  2  sin4  ^  P 
sin  1"  sin  1" 

cos  Z .  cos  d  /cos  I .  cos  d\* 

x  =  k.    .    n      ^  —m.cot(l—d).(    .  I        .  (220) 

sm  (I  —  d)  \  sm  (/  —  d)  / 

§  699.  Now  P  is  to  be  found  from  the  time  when  the  stai  is  on  the 
meridian  and  that  of  observation,  being  equal  to  the  difference  of  the  two 
converted  into  arc.  These  times  are  to  be  taken  from  a  time-piece,  and 
this  never  runs  accurately  to  sidereal  or  mean  solar  time.  If  the  time- 
keeper run  too  slow,  the  difference  of  its  indications  would  be  less  than 
the  corresponding  difference  of  true  hour  angles — if  too  fast,  the  contrary  ; 
and  P,  in  the  formula,  must  be  corrected. 

Lot  the  time-piece  lose  r  seconds  a  day  ;  then  while  the  true  day  will  be 
equal  to  86400",  the  clock  indication  will  be  86400s  —  r,  and  any  two 
corresponding  hour  angles,  one  being  the  true  and  the  other  that  indicated 
by  the  time-keeper,  denoted  respectively  by  P  and  P',  will  bear  the  re- 
lation 

P  :  P'  :  :  86400  :  86400  —  r ; 
whence 

86400  1 

'  86400  —  r  ~        '          ~       ' 

"  86400 
making 

=  0.000011  .r 


86400 
developing  the  fraction,  and  neglecting  the  higher  powers  of  r'y 

P  =  Pf  (1  +  r')  =  P'  +  P'  rr, 
and 

sin  JP  =  sin  J  P'  cos  \  r> P'  +  cos  \P>  sin  \r  P' ; 

making  cos  -J  r'  P  =  1,  squaring  and  rejecting  the  term  containing  the 
second  power  of  sin  J  r'  P',  we  find 


TERRESTRIAL    LATITUDE.  195 

sina  JP  =  sin2  £  P'  +  2  sin  \  P'  cos  i  P' .  sin  \  r'  P'  ; 
but 

2  sin -JP'.  cos  JP'  =  sin  P', 

and  since  P'  and  r'  are  both  small, 

sin  P'  =  2  sin  £  P', 

sin  i  r'  P'  =  rf  sin  1  P7  ; 

which  substituted  above  give 

sin2  £  P  =  sin2  A  P'  +  2  r'  sin2  J-  P'  =  (1  +  2  r')  sin2  £  P' ; 

and  finally  making 

i  =  1  +  2  r'  =  1  +  0.000022  r       .     .     .     .     (222) 

and  substituting  in  Eq.  (220)  we  have 

,    cos  I .  cos  d  /cos  / .  cos  dV 

x  =  i.Jc.— — =r »2.w.cot  (l—d)  .(-7—77 7-1       (223) 

sin  (l—d)  \siu(l  —  d)J 

m  which  it  will  be  recollected  that  r,  in  the  value  of  «',  is  the  rate  of  the 
time-keeper,  minus  when  the  latter  gains  and  plus  when  it  loses  on 'side- 
real time. 

§  700.  The  first  term  in  the  second  member  of  Eq.  (223)  will  always 
">e  sufficient  when  the  observations  are  made  within  five  or  ten  minutes  of 
.he  meridian.  And  it  is  important  to  remark,  in  view  of  the  use  presently 
t,>  be  made  of  the  value  of  X9  that  the  latter  will  not  be  sensibly  affected 
by  a  small  error  in  the  value  of  /,  and  that  an  approximate  latitude  may 
therefore  be  substituted  therefor.  The  values  of  k  and  m  are  computed  for 
all  values  of  P'from  0  to  35m,  and  inserted  in  Tables  V.  and  VI. 

TERRESTRIAL  LATITUDE  AND  LONGITUDE. 

§  701.  The  determinations  of  terrestrial  latitude  and  longitude  by 
,  means  of  astronomical  observations  and  ephemerides,  are  among  the 
most  important  of  the  objects  of  practical  astronomy.  All  appreciate 
the  value  of  these  determinations  in  navigation  and  geography,  and  we 
now  proceed  to  consider  them  in  the  order  named. 

Terrestrial  Latitude. 

§  702.  The  zenith  distance  of  the  pole  is  always  the  complement  of  the 
latitude  of  the  place,  and  when  known  the  latitude  is  known  from  the 
relation 

X  ^  90°  —  L 


106  SPHERICAL    ASTRONOMY. 

in  which  X  denotes   the   zenith   distance  "g'_ 

of    the   pole,   and    I   the   latitude   of  the 

place. 

§  703.  The  zenith  distance  of  the  pole 
forms  one  side  Z  P  of  a  spherical  triangle, 
of  which  the  two  other  sides,  Z  S  and 
P  S,  form,  respectively,  the  zenith  and 
polar  distances  of  some  heavenly  body, 
of  which  the  angle  at  the  pole  is  the 
hour  angle,  or  distance  of  the  body  from 
the  meridian.  And  the  determination  of  latitude  consists  in  the  solu- 
tion of  this  triangle,  the  data  for  this  purpose  being  the  true  zenith 
distance  Z  S  determined  from  observation,  the  polar  distance  P  S  found 
from  the  ephemeris,  and  the  hour  angle  Z  P  S,  which  is  always  equal  to 
the  sidereal  time  of  observation,  diminished  by  the  body's  right  ascension 
at  the  same  instant.  Having,  then,  found  the  true  zenith  distance  by  cor- 
recting the  observed  for  refraction,  parallax,  and  semi-diameter  when  ne- 
cessary, and  the  body's  true  hour  angle  and  polar  distance  from  the  time 
of  observation,  the  ordinary  formulas  for  the  solution  of  spherical  triangles 
will  do  the  rest. 

§  704.  Latitude  by  Meridian  Zenith  Distance  of  a  Body. — But  it  is 
desirable,  in  practice,  to  select  those  moments  for  observations  which  will 
give  most  accurate  results,  and  these  are  when  the  hour  angle  is  0°  or 
180°  ;  in  other  words,  when  the  body  is  on  or  near  the  meridian,  for  then 
it  has  the  least  change  in  zenith  distance  for  a  given  interval  of  time. 

Make 

z  =  Z  S  =  true  zenith  distance  of  body ; 

d  =  90°  —  P  S  =  the  body's  declination ; 
P  =  Z  P  S         —  hour  angle  of  the  body ; 
A  =  P  Z  S  —  180°—  the  body's  azimuthal  angle. 

Then  in  the  triangle  Z  P  S, 

cos  2  =  sin  /  .  sin  d  -f-  cos  /  .  cos  d  .  cos  P  .     .     .     (224) 
sin  d  =  sin  I .  cos  0  -f-  cos  /  .  sin  z  .  cos  A  .     .     .     (225) 

§  705.  Making  P  =  0°,  the  body  will  be  on  the  meridian  some- 
where between  the  poles  on  the  side  of  the  zenith,  and  A  will  be  0° 
or  180°. 

In  the  first  case,  the  body  will  be  between  the  zenith  and  elevated  role 
cos  A  —  I,  and  Eq.  (225)  will  become 


wuence 
and 


TERRESTRIAL   LATITUDE.  197 

sin  d  =  sin  / .  cos  z  +  cos  I .  sin  z  =  sin  (/  -f  z) , 
«*«*  +  «, 
J  =  d  — z (226) 

Flf.  118.  Fig.  114. 


Iii  the  second  case,  the  body  will  be  on  the  opposite  side  of  the  zenith 
from  the  elevated  pole,  cos  A  =  —  1;  and  if  the  latitude  and  declination 
be  of  the  same  name,  sin  d  and  sin  I  will  have  the  same  sign,  and  Eq.  (225) 
gives 

gin  d  =  sin  /  .  cos  z  —  cos  / „  sin  z  =  sin  (/  —  z) ; 
whence 

d  =  /  -  z, 
and 

2  =  d  +  z (227) 


Fig.  115. 


If,  in  the  second  case,  the  declination 
and  latitude  be  not  of  same  name,  the 
body  will  be  below  the  equinoctial ;  sin  d 
and  sin  /  will  have  contrary  signs,  and 
Eq.  (225)  gives 


whence 
and 


sin  (—  d)  =  sin  I  .  cos  z  —  cos  I  .  sin  z  —  sin  (/  —  z); 


l-t-d 


(228) 


198 


SPHERICAL   ASTRONOMY. 


If  P  =  180°,  the  body  will  be  on  the 
meridian  below  the  elevated  pole,  and 
A  =  0°  ;  cos  P  =  -  1,  and,  Eq.  (224), 

cos  «=  sin  /.  .  sin  d  —  cos  I  .  cos  e?=  —  cos  (/-f-d); 
whence 


Fig.im 


and 


1=  180°  -z  +  d 


(229) 


§  706.    Latitude   by   Circum-meridian 
Altitudes. — Thus  it  is  easy  to   find   the 

latitude  when  the  meridian  zenith  distance  and  declination  of  a  heav 
enly  body  are  known.  The  declination  is  found  from  the  ephemeris, 
if  the  body  belong  to  the  solar  system,  or  from  the  catalogue,  if  it  be  a 
star.  The  meridian  zenith  distance  is  best  determined  by  the  method  of 
circum-meridian  altitudes,  which  consists  in  measuring  with  an  instrument 
•a  number  of  altitudes  of  the  body  just  before  and  after  its  meridian  pas- 
sage, noting  the  corresponding  times ;  reducing  to  the  meridian,  taking  an 
average  value  of  the  results,  and  subtracting  this  from  90°. 

§  707.  Denote  by  A,,  A2,  A3,  &c.,  the  measured  altitudes ;  r},  rs,  r3,  &c., 
the  corresponding  refractions;  ph  p.2,  p3,  &c.,  the  parallaxes;  A  the  ap- 
parent semi-diameter ;  xlt  xa,  x3,  &c.,  the  reductions  to  the  meridian  ;  n  the 
number  of  observations ;  and  IT  the  average  meridian  altitude;  then  will 


—  r,  -f  fft 


-f  &c. 


(230) 


the  upper  sign  corresponding  to  the  lower  limb,  and  vice  versa.  Denote 
by  P,,  P2,  P3,  &c.,  the  watch  hour  angler  of  the  body  ;  that  is,  the  differ- 
ence between  the  watch  time  of  meridian  passage  and  those  of  observa- 
tions. These,  with  tables,  give  &„  &2,  &3,  &c.,  m,,  ra2,  ra3,  <fec.,  Eq.  (223); 
and  making 


2  m  =  m 
2  A  =  h, 
2  a:  =  or, 


m2 


&c.  ; 
?  <kc.  ; 
&c.; 


Sr 
w 


Sp 

-^ 

n 


cos  I  .  cos  c? 

-r 
am 


S  wi 


cos  ^  .  cos  d\  2 


(231) 

But  this  supposes  I  to  be  known.     An  approximate  value  will,  §  700, 
be  sufficient  ;  and  to  obtain  it,  correct  the  altitude  nearest  the  meridian  for 


TERRESTRIAL    LATITUDE.  199 

refraction,  parallax,  and  semi-diameter  ;  subtract  the  result  from  90°,  and 
substitute  the  remainder  for  z  in  one  of  the  equations  (226)  to  (229)  in- 
clusive, according  to  the  case. 

§  708.  Latitude  by  opposite  and  nearly  equal  Meridian  Zenith  Dis- 
tances. —  With  an  approximate  latitude,  select  one  or  more  pairs  of  stars, 
of  which  the  individuals  of  each  pair  shall  pass  the  meridian  on  opposite 
sides  of  the  zenith,  and  at  nearly  equal  distances.  Then,  preserving  the 
notation  of  §  704,  writing  the  subscripts  1  and  2  to  distinguish  the  stare, 
and  supposing  the  declinations  to  be  of  the  same  name  as  the  latitude,  we 
have,  equations  (226)  and  (227), 


and,  by  addition, 

_  di  +  d2      zl 


Denoting  by  £,  and  £2  the  observed  zenith  distances,  and  by  r,  and  r,  th« 
corresponding  refractions,  we  have 


which,  substituted  above,  give 

-*t 

If  £,  =  £g,  then  will  r,  —  rz  =  0,  and  we  have 

z  =  rfi_+rf. (233) 

and  thus  the  determination  of  latitude  will  be  made  independent  of  refrac- 
tion, which  is  one  of  the  greatest  sources  of  difficulty  in  practical  astronomy. 

If  £,  be  not  equal  to  %z,  but  nearly  so,  the  result  may  be  regarded  as 
equally  accurate,  since  the  difference  of  refraction  will  then  be  employed, 
which,  being  very  small,  will  be  sensibly  free  from  error. 

g  709.  This  simple  and  elegant  method,  which  is  one  of  the  most  accu- 
rate, and  now  very  generally  used,  was  first  employed  by  Oapt.  Andrew 
Talcott,  late  of  the  U.  S.  Engineers.  The  measurements  were  made  by 
means  of  a  zenith  telescope,  turning  about  a  vertical  axis,  and  provided 
with  a  micrometer.  The  stars  were  so  selected  that  when  one  was  brought 
within  the  field  of  view,  and  made  to  thread  the  micrometer  wire  as  it 
passed  the  meridian,  the  other  would  enter  the  field  on  turning  the  instru- 
ment 180°  in  azimuth.  The  second  star  being  made  to  thread  the  wire 


200  SPHERICAL    ASTRONOMY. 

by  the  micrometer  motion,  the  extent  of  the  latter  was  noted,  and  gave  the 
value  of  £|  —  £2.  The  value  of  r{  —  rz  was  found,  of  course,  from  the  re- 
fraction tables. 

§  710.  Latitude  by  Polaris  off  the  Meridian. — The  last  method  we 
shall  give  is  that  by  Professor  Littrow.  It  consists  in  observing  the  alti- 
tude of  Polaris  out  of  the  meridian,  and  therefore  at  any  convenient  time, 
and  reducing,  not  to  the  meridian  only,  but  to  the  pole  also ;  the  data  for 
this  purpose  being  the  star's  polar  distance,  its  true  altitude,  and  corres- 
ponding hour  angle. 

Let  Z  be  the  zenith,  P  the  pole,  S  Fig.  m 

the  place  of  the  star  in  its  diurnal  path 
S  Sf  m,  Z  S  the  arc  of  a  vertical  circle. 
Make 

/  =  latitude  ==  altitude  of  pole  = 

90°-  ZP\ 
h  =  true  altitude  of  star  =  90°  —  Z  S ; 
P  =  Z  P  S  ~  hour  angle  of  star ; 
4  =  reduction  to  the  pole  =  Z  P  —  Z  S ; 
A  =  P  S  =  star's  polar  distance. 

Then 

*«*-*; 

and 

i  =  h-4,-, 

so  that  the  latitude  is  known  when  -^  is  known. 
In  the  triangle  Z  P  S,  we  have 

cos  Z  S  =  cos  P  S  .  cos  Z  P  +  sin  P  S  .  sin  Z  P  .  cos  P; 
and  replacing  the  sides  by  their  values  in  terms  of  A,  h,  and  /,  or  h  —  >]>. 

sin  h  =  cos  A  .  sin  (h  —  40  +  s^n  A  •  cos  (^  ~~  40  •  cos  P » 
dividing  by  sin  h  and  factoring, 
1  =r  cos ^  .  (cos  A  -J-  sin  A  cot  h  .  cos  P)  —  sin  <//  .  (cos  A  .  cot  h  —  sin  A  .  cos  P). 

Make 

a  -=  cos  A  +  sin  A  .  cot  h  .  cos  P,    )  (2341 

b  =  cos  A  .  cot  h  —  sin  A  .  cos  P ;   f 

and  tht  above  may  be  written, 

1  =  a  cos  4/  —  b  .  sin  ^  .     .     .     .     .     .     (235) 

Now,  A  is  a  small  angle,  not  more  than  1°  40' ;  and  replacing  cos  A  and 


TERRESTRIAL    LATITUDE.  201 

sin  A  by  their  values  in  terms  of  A,  equations  (234)  become,  omitting  the 
powers  of  A  above  the  third, 

a  =  1  —  J  A2  +  (A  —  i  A3)  .  cot  h  .  cos  P, 
b  =  (1  —  J  A2)  .  cot  h  -  (A  -  -1  A3)  cos  P. 
Let 

<7A3+,  &c  .....     (236) 


be  the  development  of  -^  according  to  the  ascending  powers  of  A,  in  which 
there  can  be  no  independent  term  ;  since,  when  A  =  0,  then  will  4/  =  0. 

Whence 

cos  >L  =  l  -i^A2-^^3, 

sin  4,  =  A  A  -f  B  A2  +  (C-  l^3)  A8. 

Substituting  the  values  of  a,  6,  cos  4^,  and  sin  4/,  in  Eq.  (235),  we  have  the 
identical  equations, 

cot  h  .  cos  P  —  A  .  cot  h  =  0, 
-%(l+A*)  +  A  cos  P  —  ,8  cot  h  =  0, 
i  A  -  %  (  1  +  3  A*)  cos  P  -  (  C  -  |  ^43)  =  0. 
Whence 

A  =  cos  P  ; 

^  =  —  %  sin  2P  .  tan  h  ; 

(7  =  i  cos  P  .  sin  2P  ; 

which  in  Eq.  (236)  give 

4,  =  A  .  cos  P  —  I  sin  2P  .  tan  h  .  A2  +  £  cos  P  .  sin  IP  .  A8. 

To  express  -^  and  A  in  seconds,  write  -^  siQ  1"  f°r  4'  an(^  ^  sm  1"  f°r 
A,  and  make 

m  =  %  sin  1",  tt  =  £sin8l", 
then  will 

4,  =  A  cos  P  —  m  (A  .  sin  P)2  .  tan  h  +  n  .  (A  .  cos  P)  .  (A  .  sin  P)8  (237) 

This  value  applied  with  its  proper  sign  to  the  observed  altitude,  cor- 
rected for  refraction,  will  give  the  latitude.  It  is  best  to  take  some  half 
dozen  altitudes,  and  to  note  the  corresponding  times'  in  pretty  rapid  suc- 
cession ;  a  mean  of  the  altitudes  corrected  for  refraction  will  give  A,  and  a 
mean  of  the  sidereal  times  diminished  by  the  right  ascension  of  the  star, 
and  the  remainder  multiplied  by  15,  will  give  P. 

§  711.  This  method  is  of  such  practical  utility  as  to  have  caused  the 
insertion  into  *he  English  Astronomical  Ephemeris  and  Nautical  Almanac 
of  three  tables,  of  which  the  first  contains  the  value  of  A  cos  P  for  every 
10  minutes,  sidereal  time,  for  a  mean  and  constant  value  of  A;  the  second 
contains  the  values  of  —  m  .  (A  .  sin  P)2.  tan  h  ;  and  the  third  contains 


202  SPHERICAL    ASTRONOMY. 

corrections  to  be  applied  to  the  values  in  the  second  tible.  The  secona 
and  third  tables  are  arranged  in  the  form  of  double  entry,  the  arguments 
for  the  former  being  the  sidereal  time  and  altitude,  and  in  the  latter  side- 
real time  and  date. 

The  third  term  of  Eq.  (237)  is  neglected  as  being  insignificant. 

Longitude. 

§  712.  The  longitude  of  a  place  is  the  angle  made  by  its  meridian  with 
some  assumed  meridian  taken  as  an  origin  of  reference.  The  problem  ot 
longitude  is  much  more  complex  than  that  of  latitude,  and  its  solution 
consists,  as  we  have  seen,  §  94,  in  finding  the  difference  of  local  times 
that  exist  simultaneously  on  the  required  and  first  meridian. 

§  713.  Longitude  by  Chronometers. — Could  the  motion  of  a  time-piece 
be  made  perfectly  uniform,  and  the  angular  velocity  of  its  hour-hand  equal 
to  that  of  the  earth's  axial  rotation,  without  the  risk  of  variation,  the  de- 
termination of  longitude  would  be  a  simple  matter.  It  would  then  only 
be  necessary  to  put  the  time-keeper  in  motion ;  on  a  given  meridian  ascer- 
tain, by  the  methods  explained,  its  error  on  the  local  time  of  this  meridian  ; 
transport  it  to  the  unknown  meridian,  determine  its  error  on  local  time 
there,  and  take  the  difference  of  these  errors ;  this  difference  would  be  the 
difference  of  longitude  of  the  meridians  in  time. 

But  such  time-pieces  cannot  be  made.  The  results  to  which  they  would 
lead  may,  however,  be  approached  within  limits  all-sufficient  for  practical 
purposes.  It  is  only  necessary  that  the  time-keeper  shall  run  uniformly, 
a  condition  which  chronometers  have  been  made  so  nearly  to  attain  as  to 
vary  their  rate  but  half  a  second  in  31536000  seconds. 

§  714.  By  daily  observations  find  the  error  of  a  chronometer  ;  from  the 
variation  of  the  error  during  the  intervals  between  the  observations,  find 
that  for  24  chronometer  hours.  This  will  be  the  rate.  Make 

e=  error  on  local  time  on  gwen  meridian,  at  some  given  epoch; 

plus  when  too  slow,  minus  when  too  fast ; 

e  =  error  on  local  time  on  required  meridian,  at  some  subsequent  epoch  ; 
e  t  =.  error  on  local  time  on  given  meridian,  at  this  hist  epoch  ; 
r  —  rate;  minus  when  gaining,  plus  when  losing; 
i  =  interval  of  chronometer  time  between  the  .epochs  at  which  e  and 

ef  are  found — always  plus ; 
I  =  difference  of  longitude. 
Then  /  =  «-«;         .  ./ 


TERRESTRIAL    LONGITUDE. 


203 


Fig.  118. 


whence  /  =  e,  —  e  +  i  .  r (238) 

§  715.  Longitude  by  Lunar  Distances. — The  moon  has  a  rapid  motion 
in  longitude.  Her  geocentric  angular  distances  from  the  sun,  planets,  and 
fixed  stars  that  lie  in  and  about  her  path  through  the  heavens,  are  com- 
puted in  advance  and  inserted  into  the  Nautical  Almanac.  From  these 
hours  and  distances  is  readily  found,  by  interpolation,  the  Greenwich  time 
corresponding  to  any  given  distance  not  in  the  Almanac,  and  the  difference 
between  this  interpolated  time  and  the  local  time  on  any  other  meridian 
at  which  the  moon  is  found  from  observation  to  have  this  given  distance, 
is  the  longitude  of  the  meridian  on  which  the  observation  is  made. 

§  716.  Measure  the  altitude  of  the  star,  and  that  of  the  upper  or  lower 
bright  limb  of  the  moon  ;  also  measure  the  angular  distance  from  the  star 
to  the  bright  limb  of  the  moon,  and  note  the  local  time  of  this  measure- 
ment ;  correct  the  altitude  of  the  limb  and  measured  distance  for  semi- 
diameter  ;  then  correct  the  altitude  of  the  star  for  refraction,  and  that  of 
the  moon  for  refraction  and  parallax. 

Let  Z  be  the  zenith,  Z  S  and  Z  M  the  arcs 
of  vertical  circles,  the  first  passing  through  the 
star  S  and  the  second  through  the  moon's  cen- 
tre M.  The  effect  of  refraction  being  to  ele- 
vate and  that  of  parallax  to  depress,  and  the 
parallax  of  the  moon  being  always  greater  than 
her  refraction,  the  star  will  appear  at  S'  above 
its  true  place,  and  the  moon  at  Mf  below  her 
true  place. 

Make 

h  —  90°  —  Z  M'  =  observed  altitude  of   moon's    limb  corrected  for 

semi-diameter ; 

h'  —  90°  —  Z  S'  =  observed  altitude  of  star ; 
dr  =  M' Sr  =  observed  distance  corrected  for  semi-diameter  of 

the  moon ; 

H  =  90°  —  Z  M  =  true  altitude  of  moon's  centre  ; 
H'  =  90°  —  Z  S   =  true  altitude  of  star ; 
4  =  M  S  =  true  or  geocentric  distance  between  the  moon's 

centre  and  the  star ; 
z  =  MZS          =  angle  at  Z. 

Then  in  the  triangle  M'Z  S', 

cos  A'  —  sin  h  .  sin  h' 


cos  z  = 


cos  h  ,  2os  h' 


2Q4-  SPHERICAL    ASTRONOMY. 

and  in  triangle  M  Z  S, 

cos  A  —  sin  H .  sin  H' 

cos  z  — —= ~, ; 

cos  H .  cos  If' 

equating  these  values  of  cos  z, 

cos  A'  —  sin  h  .  sin  h1       cos  A  —  sin  H .  sin  IT 
cos  h  .  cos  h'  cos  H .  cos  H' 

adding  unity  to  both  members  and  reducing, 

cos  A'  +  cos  (h  +  h')  _  cos  A  +  cos  (H  +  H')   ^ 
cos  h  .  cos  h'  cos  If .  cos  If' 

Make 

A  +  A'  +  A'  =  2  ra (239; 

whence 

cos  (h  -h  h1)  =  cos  (2  m  —  A') ; 

substituting  this  above  and  reducing,  we  find 

,   H+H'  2A 

,  ,.        cosj . sin2  — 

cos  m  .  cos  (m  —  A  )  2  2 

cos  h .  cos  A'  cos  H .  cos  ^P 

whence 

sin  1  A  =  '/cos2  J  (J5T  +  £T)  -  cos^-cosfr  .  cos  m  .  cos  (m  -  A'), 

'        cos  h  .  cos  A' 

and  making,  to  adapt  the  foregoing  to  logarithmic  computation, 


.  COS 


— : rr  •  cos  m  .  cos  (m  —  A') 

cos  A  .  cos  A 

.     .     (240) 


then  will  result 

sin  l  A  =  cos  i  (H  +  If')  .  cos  9      .     .     .     .     (241) 

§  717.  The  quantities  A,  A',  H,  H',  and  A',  are  obtained  from  observa- 
tions, and  the  corrections  for  semi-diameter,  refraction,  and  parallax  applied 
thereto  ;  the  value  of  m  is  given  by  Eq.  (239)  ;  the  auxiliary  arc  <p  by 
Eq.  (240),  and,  finally,  the  true  distance  A  by  Eq.  (241.)  This  operation 
is  technically  called  clearing  the  distance. 

§  718.  With  this  distance  enter  the  Nautical  Almanac  and  see  if  it  is 
found  therein  ;  if  it  is,  take  the  corresponding  time  from  the  head  of  the 
column,  and  subtract  therefrom  the  local  time  of  observation  ;  the  remain- 
der will  be  the  longitude  —  west  if  this  remainder  be  plus,  east  if  it  be 
aegative. 

§  719.  If  the  precise  distance  be  not  found  in  the  Almanac,  as  it  sel 


TERRESTRIAL   LONGITUDE.  205 

dom  will,  find  two  consecutive  distances,  one  of  which  is  greater  and  the 
other  less.  Take  these  and  the  next  two  or  more  precedirg  and  following 
distances,  and  form  their  first,  second,  third,  &c.,  differences,  denoted  re- 
spectively by  Ab  A2,  A3,  &c.,  in  which  A,  is  the  difference  between  the 
consecutive  distances  of  which  one  is  less  and  the  other  greater  than  the 
given  distance.  Make 

I)  =  given  distance  ; 
I)'  =  nearest  distance  in  ephemeris  ; 
T'  =  ephemeris  time  corresponding  to  Dr, 
T  =  Greenwich  time  corresponding  to  D  : 
t  =  T  —  T. 

Then  because  the  ephemeris  intervals  are  3h,  will,  by  the  ordinary  for- 
mula for  interpolation, 


supposing  the  second  differences  constant,  which  we  may  do  without  sen- 
sible error,  and  solving  with  respect  to  first  power  of  /, 


Neglecting  the  second  difference,  we  have 

D  —  Dr 
*  =  -^—  3"     .......     (2*3) 

which  in  the  denominator  of  the  preceding  equation  gives 

7)  —  D' 
t=  -  --  -8*.     .     .     •     (244) 

A    -      A 


.    .    .     (245) 


and  replacing  t  by  its  value  T  —  T  ',  we  have  finally 

D  —  D' 


A  - 


§  720.  A  single  observer  begins  by  taking  with  his  sextant  an  altitude 
of  the  star,  then  an  altitude  of  the  moon's  bright  limb,  then  the  distance  be- 
tween the  star  and  moon's  limb,  then  the  altitude  of  the  moon's  bright  limb, 
then  the  altitude  of  star,  carefully  noting  the  time  of  taking  the  distance. 
A  mean  of  the  altitudes  of  the  moon  and  star  will  give  the  approximate 
altitudes  which  the  moon  and  star  had  when  the  distance  was  measured.  - 


206  SPHERICAL     ASTRONOMY. 

§  721.  It  is  scarcely  necessary  to  add,  that  if  the  sun  or  a  planet  be 
taken  instead  of  a  star,  corrections  for  semi-diameter  and  parallax  must 
be  added  to  that  of  refraction. 

§  722.  Longitude  by  Lunar  Culminations.  —  If  the  change  in  right 
ascension  of  a  point  of  the  moon,  in  its  passage  from  one  meridian  to 
another,  be  known,  the  distance  between  the  meridians  becomes  known 
from  the  point's  rate  of  motion  in  right  ascension.  Make 

cl  =  the  point's  right  ascension  when  on  any  upper  first  meridian  ; 

c,  —  its  right  ascension  when  on  an  upper  known  meridian  to  the  west. 

H  •=.  longitude  of  this  known  meridian,  west. 

/  =  approximate  longitude  of  any  unknown  meridian  between  these. 

L  =  true  longitude  of  the  same. 

e  =  L  -  1 

a  =z  right  ascension  of  the  point  when  on  the  upper  meridian,  of 
which  the  longitude  is  L. 

§  723.  —  1st  Approximation.  Then,  were  point's  motion  in  right 
ascension  uniform, 

Cg  —  Cj  :  //  :  :  a  —  c,   :  I 

or  H      . 

I  =  —-  (a  -  Cl) 

Cg  -  Cl 

§  724.  —  %d  Approximation.  But  the  moon's  motion  in  right  ascen- 
sion is  not  uniform,  and  the  above  will  in  general  be  erroneous,  and  by 
the  quantity  e,  which  is  a  small  arc  of  longitude  ;  and  we  have 


The  arc  e  being  small,   the  moon's   motion  in  right  ascension  will  be 

sensibly  uniform  while  between  the  meridians,  through  its  extremities. 

Make 
at  =  the  lunar  point's  right  ascension  when  on  the  meridian,  of  which 

the  longitude  is  £, 

v  =  the  point's  rate  of  motion  in  right  ascension  ;  and  let  this  be 
measured  by  the  distance  in  right  ascension  over  which  the 
point  would  move,  with  this  rate  constant,  while  between 
the  meridians  of  which  the  distance  apart  is  H. 

Then  by  the  principle  above,  writing  e  for  /,  v  for  c2  --  c1?  and  aA  for  cw 

we  have 


and  this  in  the  abov  3  gives 


TERRESTRIAL     LONGITUDE. 


207 


§  725.  Now,  these  equations  will  be  equally  true  from  whatever 
point  of  the  equinoctial,  taken  as  an  origin,  the  right  ascension  be 
estimated.  For  convenience,  take  the  origin  at  the  declination  circle 
through  the  lunar  point  at  its  last  passage  over  the  first,  or  meridian  of 
the  Ephemeris.  Then  will 


c2  =  change  of  right  ascension  between  the  known 

meridians, 
a  =  increase  of  right  ascension  from  the  first  to 

the  intermediate  or  required  meridian, 
04  =  increase  of  right  ascension  from  the  first  to 

the  approximate  meridian  I. 

With  this  new  notation  the  above  equations  become 


(246) 


(247) 


§  726.  In  the  Nautical  Almanac  and  Astronomical  Ephemeris  are 
given  the  right  ascension  of  the  point  of  the  bright  limb  at  which  a 
declination  circle  is  tangent  to  the  lunar  disc,  and  also  the  right  ascen- 
sions of  one  or  more  stars,  at  the  instant  of  passing  the  upper  and  lower 
meridian  of  Greenwich  for  every  day  in  the  year.  The  stars  are  so 
situated  as  to  lie  about  the  moon's  parallel  of  declination,  and  not  far 
from  her  in  right  ascension. 

§  727. — 1.  Interpolation.     Take  the  following  scheme: 


I 

F 

AI 

A, 

A3 

A4 

A9 

*'" 

a'" 

t" 

a" 

V* 

c" 

V 

df 

t' 

a' 

cr 

e' 

tl 

a, 

b 

c, 

d 

e{ 

* 

i, 

*„ 

*„ 

*„ 

t,., 

a 

\ 

in  which  the  column  I  contains  the  independent  variable,  or  argument, 
as  time,  terrestrial  longitude,  degrees,  and  the  like ;  F  the  value  of  a 


208 


SPHERICAL     ASTRONOMY. 


function  of  this  variable,  as  found  in  any  set  of  tables  ;  A1?  A2,  A3,  etc., 
the  first,  second,  third,  etc.,  orders  of  differences  of  these  functions. 

Make 

*  =  the  interpolated  value  of  the  function  correspond- 
ing to  any  given  value  tt  of  the  argument  be- 
tween t'  and  tt ; 


t,  —  t' 

A,  =  6, 


2     ' 


A3  =  d, 


e'  +  e, 


(248) 


Then,  limiting  the  operation  to  the  fourth  order  of  differences,  will 

*  =  a'  +  A  t  +  BP  +  Ct*  +  D  t*  + 
a,  =  s  -  a'  =  At  +  BP  +  Of   \-  D  P        (249) 
in  which 


A  rr  Aj  -  1 


.    (250) 


Also  taking  first  differential  coefficient  of  the  function  (249) 

v  =  A  +  2Bt  +  SCt2  +  4Z>*3   ....    (251) 

which  would  be  the  increment  of  the  function  for  an  increment  ot  t 
equal  to  unity,  were  the  function  to  increase  uniformly  and  at  the  rate 
it  had  for  any  arbitrary  value  for  tt. 

§  728.— 2.    Observations.'    Make 

.<?&  =  sidereal  time  of  moon's  bright  limb  passing  meridian. 

h$  —  clock  time  of  same  passing  line  of  coliraation. 

e$  =  clock  error  at  same  instant. 

i$  =  error  of  transit  for  altitude  of  moon,  in  time  seconds. 

Then,  §§211  and  729, 


TERRESTRIAL    LONGITUDE.  209 


=  A,  +  e,  ±  —  (1  T  coe  /.  sec  D  ,  ?  .  sin  P); 


the  upper  sign  before  meridian  passage,  and  in  which  I  is  the  latitude 
of  the  observer,  p  the  radius  of  the  earth  at  his  place,  D  the  moon's 
declination,  P  her  equatorial  horizontal  parallax,  and  m  her  daily  mo- 
tion in  right  ascension,  in  hours. 
Making  similar  notation  for  a  star, 


subtracting  this  from  the  preceding,  and  writing  <p  for  e$  —  e%,  the 
clock  acceleration  in  the  interval,  in  time,  between  the  moon  and  star, 


On  a  second  meridian,  to  the  west,  a  similar  equation  is  found,  with  the 
variables  accented.     Taking  the  difference,  and  making 


there  will  result 

^  =  s'5  —  s9  =  (A'  5—  A'*  4-  9')  —  (A&-  A*  4-  <p)±  &2     .     .     (252) 

Or,  if  there  be  but  one  observer  with  Ephemeris,  then  will  i$  =  0,  and, 
omitting  accents,  the  value  of  k^  becomes, 

k*  =  =*=  i-o,04  166m  '  (°'°41  66  •  m  =F  sec  ^  •  cos  l  •  P  •  sin  P)- 
?3,  is  found  by  the  method  of  13,  Appendix  II.,  p.  261. 

in  which  A  denotes  the  same  as  a,  in  Eq.  (246),  when  a  single  observer, 
on  an  unknown  meridian,  employs  the  ephemeris  elements,  as  given  foi 
the  next  preceding  passage  over  the  first  meridian,  with  those  of  his 
observations,  to  get  the  increase  of  right  ascension  requisite  to  find  his 
approximate  longitude  /;  and  the  same  as  a  —  ar  in  Eq.  (247),  when 
he  employs  either  the  interpolated  or  observed  increase  of  right 
ascension  for  the  meridian  of  which  the  approximate  longitude  is  £, 
to  correct  its  place.  In  the  first  case  c2  is  the  difference  of  the  point's 
right  ascension,  as  given  for  the  next  preceding  upper  and  next 
following  lower  culmination  over  the  first  meridian  ;  in  the  second  case 
v  will  be  given  by  Eq.  (251);  and  in  both,  H  will  be  12  hours  of 
longitude. 

14 


210 


SPHERICAL   ASTRONOMY. 


Example. 

OBSBKVATIONS.—  West  Point,  iS45,  Feb.  18. 
f  Geminorum    .    .    6h  54m4i»,  76 
<J          "  .    .    7    1°   38,97 

D  W.  Limb  ..........   7h  38™  o6»,  76 

£  Cancri    .    .    .    .    8   o3   06,  n 

3)22    08    26  ,  83          7    22   48  ,  94 


Jd  =  o ; Clock  rate,  4-  3s 

Nautical  Almanac, — Greenwich,  same  date. 
£  Geminorum  .  .  6h  54™  57»,  41 
<J  «  .  .  7  10  54,36 

J>  W.  Limb 7h  27"  47*,  66 

f  Cancri    .    .    .    .    8   o3    21  ,44 

3)22    09    i3  ,  21  7    23    04 ,  4o 

a  =  A  = 


fhen,  Eq.  (246), 

E  —  I2&oo»oo», 
a  =  oo    10   34  ,  53 
Nautical  Almanac    ca  =  oo    25    41  ,  18 
t  =    4    56    28, 


Log  .  .  4,6354837 
"  .  .  2  ,  8024620 
"  a.  c.  6  ,  8121918 


»      .    .    4,  2601275 
Next,  interpolate  change  of  right  ascension  for  I  ;      . 
4*>  56»  28' 


o»»  i5«  17',  8a 
—  o  ,  o3 


o   04   43  ,  26 
o'a  io">34s,  53 


/  = 


j—  , 


12   oo    oo, 

t    .    .    .    . 


Log  .  .  4,  2501275 
"  a.c.  5,3645i63 
"  .  .  9,6146436 


Nautical  Almanac. 

Feb.  17,    L.  C.    7h  oi™56',  27 

"     18,    U.  C.    7    27    47,66 

«     «      L.  C.    7    53    28  ,  84 

»     19,    U.C.    8    18    59,56 


25»5is,  39 
25  41  ,  18 
25  30,72 


(-10*,  21 

.J 

/  —  10  ,  46 


A  =  25«4i8,  18  +  o5»,  17  -  o»,  02  =  25"»  46',  33 
B-—    o5,T7-foo,o6  =—    o5,n 

C-     ......    .  -    00.04 


Then,  Eq.  (249), 


A    .    . 

Log 

.    .    3  ,  1893022 

t    .    . 

u 

.     .    9  ,  6i46438 

2  ,  8039400 

Nos    .    .         63£s,  72 

B    .    . 

Log 

.    .    o  ,  7084209 

F    .    . 

u 

.     .    9  ,  2292876 

9,9377o85 

Nos    .    .        —  o  ,  87 

0   .    . 

Log 

.    .    8  ,  6190933 

*    .    . 

« 

.    .    8,84393i4 

^  ,  4630247 

Nos    .    .       —  o  ,  oo3 

* 

....         635,85 

io»  34s,  53  =  o 

....         634,53 

Again,  Eq.  (251), 

A Nos    .    .    s5»46«,  33 

B  .    .    Log    .    .   "o ,  7084209 

t  .    .      «      .    .    9,6146438 

8  .    .      ««      .    .    o ,  3oio3oo 


o  ,  6240947        Nos 


C  .  .  Log  .  .  8,6190933 
*»  .  .  "  .  .  9 ,  2292876 
3  .  "  •  •  o,477'2'3 


8,  3255022        Nos    .    .       -  0,02 


«  =    .    .    .    .    s5°>  4a%  10 
Then,  last  term  of  Eq.  (247), 

H   .    .     Log    .    .    4,6354837 

a -a,    .    .       "       .    .    o",  1205739 

v     .    .       "     a.  c.    6,8118875 

=  -  36%  97    .    .       "       .    .    T ,  5679451        No*    .    .    oo    36 , 97 

Kq.  (247), 

L  =  4h  56m  28'  -  36«,  97  =  4h  55"«  5r,  o3 


212  SPHERICAL   ASTRONOMY. 

§  729.  It  frequently  happens  that  the  moon  cannot  be  observed  on  the 
middle  wire,  in  which  case  she  is  far  enough  from  the  meridian  to  have  a 
sensible  parallax  in  right  ascension  ;  and  as  it  may  be  very  desirable  not  to 
lose  the  observation,  this  parallax  must  be  computed  and  applied  to  the 
apparent  hour  angle  from  the  middle  wire,  which  is  supposed  to  be  nearly 
coincident  with  the  meridian. 

Denoting  the  hour  angle  by  A,  the  parallax  in  hour  angle  by  A  A,  th,-. 
geocentric  latitude  by  /,  the  moon's  declination  by  Z>,  and  her  horizontal 
parallax  by  P,  then,  Appendix  XI,  p.  379, 

A  h  —  p  .  cos  /  .  sin  P .  sin  h  .  sec  D  ; 

and^to  make  this  applicable  to  the  case  before  us,  A  will  denote  the  equa- 
torial interval,  in  sidereal  time,  from  the  lateral  to  the  central  wire.  This 
angle  being  small,  its  arc,  expressed  in  seconds  of  time,  may  be  taken  for 
its  sine,  in  which  case,  A  A  will  be  in  time-seconds,  and  the  true  distance 
of  the  moon's  limb  from  the  central  wire,  denoted  by  A/?  will  be 

h,  ==  A  .  (1  —  p  .  cos  /.sin  P .  sec  D) ; 
ind  the  reduction  to  the  meridian,  denoted  by  r,  in  time-seconds, 

A        1  —  p  .  cos  I .  sin  P .  sec  D 
cos~7>  ' ~         1  —  0,04166  .m 

in  which  m  is  the  moon's  daily  motion  in  right  ascension  in  hours. 
The  upper  sign,  when  the  observation  is  before  the  middle  wire.  The 
quantities  p  and  I  are  found  from  tables  on  pp.  336,  337. 

§  730.  It  also  often  happens  that  two  observers  do  not  use  the  same 
number  of  wires,  or  if  they  do,  that  the  same  stars  are  not  observed  at 
the  same  number.  Such  observations  are  not  of  equal  weight.  To  rind 
the  relative  value  with  which  such  observations  should  enter  into  the 
tinal  determination,  Professor  Gauss  has  given  the  following  formula, 
deduced  from  the  principle  of  least  squares. 

Let  the  number  of  wires  on  which  the  moon  is  observed  at  one  place  be 
denoted  by  nt  and  at  the  other  by  n1 ';  and  let  the  number  of  wires  at 
which  the  stars  are  observed  at  the  first  place  be  a,  b,  c,  &c.,  and  at  the 
other  be  a',  &',  c',  &  3.  Make 

r 

=  X (253) 


n  + 

_a_^_   =  a      _A^_  =  R      -^L  =  r,  fee.  (254) 

a  +  a'  b  +  \i  c  +  c 

tf  =  a  +  £  +  y  +  &r.  (255) 


TERRESTRIAL    LONGITUDE.  213 

Then,  if  W  denote  the  weight  of  each  day's  comparison,  will 


in  which  z  is  the  same  as  —      -  in  Eq.  (247)  ;  and  for  the  weight  of  the 
result  of  all  the  comparisons,  we  have 

.......  (257) 


in  which  2  expresses  the  sum. 

Let  e  denote  the  probable  error  of  observation,  and  E  the  probable  error 
of  the  final  result,  then  will 

.....  (258> 


§  731.  Longitude  by  Telegraph.  —  One  of  the  simplest  and  most 
accurate  methods  for  finding  differences  of  longitude,  is  to  telegraph  to 
a  western,  the  instant  of  a  fixed  star's  culmination  at  an  eastern  station, 
and,  conversely,  to  telegraph  to  the  eastern  the  instant  of  culmination 
of  the  same  star  at  the  western  station.  The  local  times  of  both  events 
being  noted,  the  difference,  as  recorded  at  the  same  station,  corrected 
for  rate  of  time-keeper,  gives  the  difference  of  longitude. 

The  instant  of  culmination  of  the  moon's  bright  limb  being  also  sig- 
nalized in  the  same  way,  the  difference  of  time,  as  recorded  at  the  same 
station,  corrected  for  rate,  as  before,  gives  the  difference  of  longitude 
augmented  by  the  limb's  change  in  right  ascension  during  the  interval, 
and  the  excess  of  this  interval  over  that  for  the  fixed  stars  is  the 
change  itself.  Thus  the  telegraph,  where  it  connects  stations  remote 
froir  one  another,  gives  the  means  for  finding  differences  of  longitude 
and  for  correcting  the  lunar  ephemeris,  and,  therefore,  the  elements 
employed  in  the  method  of  lunar  culminations,  for  use  at  stations 
having  no  telegraphic  connections. 

§  732.  Longitude  by  Solar  Eclipse,  or  by  Occupation.  —  The  follow- 
ing elegant  and  accurate  solution  of  this  most  important  problem  is, 
in  substance,  due  to  Mr.  Woolhouse  ;  it  first  appeared  in  the  Nautical 
Almanac  for  1837. 


214:  SPHERICAL   ASTRONOMY. 

Let  M and  S,  be  the  moon  and  sun,  in  such  geocentric 
positions  as  to  appear  in  external  tangential  contact 
to  an  observer  on  the  earth's  surface ;  the  local  time 
of  this  observer  will  be  that  of  beginning  or  ending  of 
the  local  eclipse^  Conceive  a  fictitious  sun,  s,  at  the 
distance  of  the  moon,  within  and  tangent  to  the  visual 
cone  that  projects  the  true  sun  on  the  celestial  sphere 
for  this  observer.  This  fictitious  sun  will  be  in  con- 
tact with  the  moon ;  and  any  parallactic  effect  on  the 
one,  due  to  a  change  in  the  observer's  place,  will  be 
equal  to  that  on  the  other.  Transport  the  observer 
to  the  centre  of  the  earth;  the  moon  and  fictitious  sun  will  appear  to 
shift  their  places  with  respect  to  the  true  sun;  but,  being  in  actual, 
will  remain  in  apparent  contact.  The  apparent  disk  of  the  fictitious 
sun  and  of  the  moon  will  diminish ;  and  the  size  and  place  of  the  latter 
will  become  those  of  the  ephemeris  at  the  instant  of  observation.  The 
change  of  the  fictitious  sun's  place,  in  reference  to  that  of  the  trne  sun, 
will  be  the  effect  of  relative  parallax.  Apply  this  parallax  to  the 
place  of  the  true  sun,  and  diminish  his  disk  by  a  quantity  equal  to  the 
diminution  of  the  fictitious  sun  ;  the  result  will  be  the  place  and  size 
of  the  latter  body  in  apparent  contact  with  the  moon,  to  the  observer 
at  the  central  station.  The  ephemeris  time  of  this  contact,  diminished 
by  the  local  time  of  observation,  will  give  the  longitude  of  the  observer. 

Thus,  the  determination  of  terrestrial  longitude,  by  a  solar  eclipse,  is 
reduced  to  finding  the  ephemeris  time  when  the  true  disk  of  tlie  moon 
comes  in  contact  with  a  disk  of  a  given  size,  placed  at  a  given  place. 
The  principle  is  the  same  for  an  occultation  of  a  star  by  the  moon. 

In  the  case  of  a  solar  eclipse,  the  apparent  time  of  observation, 
converted  into  arc,  gives  the  hour  angle  of  the  sun's  centre  at  that 
instant ;  and,  as  the  declination  of  the  sun  is  never  subject  to  a  very 
rapid  daily  variation,  this  element  may  be  taken  from  the  ephemeris, 
with  sufficient  accuracy,  for  the  approximate  local  time  on  the  meridian 
for  which  the  ephemeris  is  constructed,  deduced  from  an  estimated 
longitude,  or  rough  longitude,  by  account. 

Take 

a  =  right  ascension, 

h  =  hour  angle, 

}.  of  true  sun; 
6  —  declination, 

a  —  apparent  semi-diameter, 


TERRESTRIAL    LONGITUDE.  215 

Take  also 

a0  =  right  ascension,  "| 

#o  =  declination,  V  of  fictitious  sun, 

tfo  =  apparent  semi-diameter,  J 

Act  =  Ah  —  relative  parallax  of  moon  in  right  ascension, 
Ad  =       «  "  "          declination, 

Arf  =  diminution  of  fictitious  sun's  semi-diameter;. 

Then  will 

a0  =  a  -f-  &h, 

x        x  J    A*  FIg>  m 

o0  =  o  -{-  Ao, 

tf0  =  <f  —  Atf. 

Let  JV,  be  the  north  pole;  -3£  the  place  of  the 
moon  at  the  instant  of  contact;  w,  her  place  when  in 
conjunction  with  the  fictitious  sun,  s. 

Make 

( t )  =  any  convenient  ephemeris  time,  near  this  conjunction, 
(A)  =  moon's  right  ascension  at  (t), 
(D)  =       "       declination  at  (*), 
Al  =      "       relative  motion  in  right  ascension  at  (tf), 
Dl  =      "  "  "  declination  at  («), 

<0  =  time  of  true  conjunction  with  fictitious  sun,  *, 
D0  =  declination  of  the  point  m  at  this  time. 

Then,  employing  the  parenthesis  to  indicate  the  values  of  the  severaJ 
quantities  at  the  time  (*),  we  have 

(«.)  =  (»)+ A*,       t.  =  (o  +  K)7(J). 

-«1 

(«.)  =  (i)  +  A*,       D.  =  (D)  I-  M-Z-t^A. 


216  SPHERICAL    ASTRONOMY. 

Now,  make 

r  n          /*  \ 

k  =  m  8  =    J)Q—   (60), 

A  =  Ms, 
90°  4  -n  =  Mms, 


then  will 


and,  by  the  triangle  w  Jlf  s,  considered  as  plane, 


,         k  -  cos  ?j 
cos  4/  = ; 


and,  from  the  spherical  triangle  JW  M s, 


•i.       JT«  =  -rin  A 


cos 


«r  as  the  small  arcs  are  proportional  to  their  sines, 


cos 


And  the  time  required  for  the  moon  to  change  her  hour  angle  by  this 
quantity,  will  be 

MNs  _         A_    sin  (n  +  4Q 
Al  A,"     cos(D)     ' 


TERRESTRIAL    LONGITUDE. 


217 


which,  subtracted  from  £c,  will  give  the  ephemeris  time  of  observation, 
Denote  this  time  by  t,  and  we  have 


cos  (D) 


(259) 


The  longitude,  from  the  meridian  of  the  ephemeris,  is  found  by  the 
difference  between  this  time  and  that  of  observation,  previously  making 
both  apparent,  or  both  mean  time,  by  applying  the  equation  of  time ; 
and  it  will  be  west  or  east,  according  as  the  ephemeris  time  is  greater 
or  less  than  that  of  observation. 

To  find  Act,  and  A£,  take  Eqs.  (2),  Appendix  XI.,  p.  379,  and  write 
therein  Act  for  Ah,  P  for  sin  P,  A§  for  Aj},  6  for  D  and  Dl ',  unity  for 
cos  TfAh,  and  substitute  for  h  its  value  h'  —  Ah;  we  find 


Act  =  p 


cos 
-  r 
cos  6 


sm 


AS  =  p  •  P  •  (sin  I  •  cos  8  —  cos  I  •  sin  d  cos  (h  —  ^Aa 


(260) 


in  which  I  denotes  the  central  latitude ;  and,  employing  the  method  of 
solution  in  Appendix  XI,  page  381,  we  have 


n   cos  I 

AOL  —  p  •  P  • •  sm  #, 

cos  $ 

(h)  =  h  -  iA«, 
tan  6  =  cos  (h)  -  cot  /,  tan  M  = 

tan  s  =  tan  (&  +  S)  •  cos  M, 
A§  =  p  •  P  •  cos  M'  cos  s. 


tan 


(261) 


To  find  Atf,  resume  Eq.  (27),  substituting  therein  a  for  s,  rf'  for  *', 
cos  (90°  —  s)  for  cos  Z,  unity  for  cos  z ;  and  we  have 


w  —  p  •  P '  sin  s ' 


218  SPHERICAL     ASTRONOMY. 

subtracting  unity  from  both  members,  clearing  the  fraction,  writing 
P  —  if  for  P,  and  then  P'  for  p  (P  —  *),  we  have 


,  d  -  P'  -  sin  s          <f      P'        100  •  sin  s 

(f   —  (f  =  Arf  z= 


w  —  P'  -  sin  5        10     10    u  —  P'  •  sin  s 


Taking  the  average  value  of  P'  in  the  denominator,  say  57'  03",  5, 
and  p  =  1 ;  and,  expressing  tf  and  f  in  minutes,  in  which  case  w 
=  3437,  45,  we  may  write 

tf     P 


m  which  Ac1  will  be  expressed  in  seconds,  if 

100  X  60  •  sin  s 
=  3437',  45-  57',  06- sins' 

For  an  occultation  of  a  star  by  the  moon,  the  calculation  will,  in 
some  respects,  be  slightly  abridged.  The  characters  A^  and  D} 
become  the  absolute  motions  of  the  moon  in  right  ascension  and 
declination ;  the  semi-diameter  0",  and  its  diminution  Atf,  will  reduce  to 
zero;  and  the  angle  s,  which  is  only  used  to  get  Atf,  may  be  dispensed 
with;  in  which  case  it  may  be  better  to  employ  Eqs.  (260)  than 
Eq.  (261)  ;  or  Eqs.  (261)  may  be  modified  into  the  following  con- 
venient expressions,  by  eliminating  M  and  s ;  viz.  : 

h,  (k)  =  h  -  jAa, 


(262) 
tan  6  =  cos  (h)  -cot  J, 


It  will  be  useful  here  to  recapitulate  the  equations  in  a  form  suited 
to  the  facilitating  of  arithmetical  calculation,  and  separately  to  arrange 
them  for  an  eclipse  of  the  sun,  and  an  occultation  of  a  star  by  the 
moon,  to  preserve  distinctness. 


TERRESTRIAL    LONGITUDE. 


210 


I.  —  Eclipse  of  the  Sun. 

1.  With  the  longitude  by  account  find  the  corresj  onding  Greenwich 
time,  and  thence  from  the  ephemeris  take  out  the  sun's  right  ascension  a, 
declination  c>,  and  semi-diameter  tf  ;  the  horizontal  parallaxes  P,  if  ;  also 
take  out  the  moon's  declination  J)  roughly  to  the  minute. 

Reduce  the  latitude  by  the  table  on  p.  336,  and  with  p  from  the  table 
on  p.  337,  Ap.  XI,  find 


h  =  apparent  time  of  observation  reduced  into  arc. 

P 

=  P' cos  /  sin  A ;     4  h  in  min.  =  [7.92082]— ^-yr;     (h)  =  h  — 

JcosD       v  ' 

tan  &  =  cos  (h)  cot  I ;  G  =  cos  (h)  cos  I ; 

tan  g  —  tan  (b  +  #)  cos  Jlf ; 


sn 


tan  M  —  -  -  --  -  tan  (A)  ; 
cos  (6  +  6) 


check 


5  =  cos  Jf  cos  s ; 

sin  0 


'    cos(d  +  <5) 


A  a  in  time  =  [8.82391] 


cos 


M  to  be  in  the  same  semicircle  with  k. 


3.  With  s  find  the  corresponding  factor  f  in 
the  annexed  table ;  then,  using  P  and  <f  each  in 
minutes, 


and  thence 


=  [9.43537]  P. 


f 
total  or  annular 


o 
o 

10 
20 

3o 
4o 
5o 
60 
70 
80 
90 


Fact  or  /for 
diminution  of 
O's  semi-diam. 


o-s-  :  : 


1.54 
i.67  •; 


4.  In  the  hourly  ephemeris  of  the  moon,  fix  on  a  convenient  time  (t)  at 
which  the  moon's  right  ascension  is  near  to  a0,  and  for  this  time  take  out 
the  right. ascension  (^4)  in  time,  the  declination  (71),  and  their  hourly  va- 


220  SPHERICAL    ASTRONOMY. 

nations  ;  also  the  sun's  right  ascension  (a),  declination  (5),  and  t'neir  hourly 
variations.     Then, 

^4,  =  hourly  var.  (A)  —  hourly  var.  (a)  in  time; 
Z>,  —  hourly  var.  (D)  —  hourly  var.  (<5)  in  arc  ; 

K)  =  (a)  +  Aa; 

(S  0)  =($)  +  A  6. 

m  =  (a°)  ~  (  J)  ;  t0  =  (t)  +  m  [3.55630]  ; 

-™i 


n  =  [1.17609]^,  cos(D); 

D\  k  cos  7) 

tan  ->j  =  --  ;  cos  •!>  =  -  . 

n  A 

Corresponding  Greenwich  mean  time  =  t0  +  [3.55  C30]  -  sin 
«  to  have  a  different  sign  from  D^  : 


upper 
under 


!(  immersion  )  . 
sign  when  an  <  V  is  observed. 

(  emersion     j 


II. —  Occupation  of  a  Star  by  the  Moon. 

6.  With  the  estimated  longitude  find  the  corresponding  Greenwich 
and  thence  take  out  the  moon's  horizontal  parallax  P,  and  her  declination 
Z/,  roughly  to  the  minute ;  also, 

sid.  time  =  apparent  time  +  O's  right  ascension ;  or, 

sid.  time  =  mean  time  -f-  sid.  time  mean  noon,  from  p.  III.  of  ephemens  • 

+  accel.  on  Greenwich  mean  time ; 
h  —  sid.  time  —  a,  in  arc ; 
P'  =  ?P', 
a  being  the  star's  right  ascension. 

'  p=P' cosl&mh',     Akm  min.  =  [7.92082]-^— ;     (h)  =  h  —  A  A; 
K  =  P'  sin  I  cos  S ;      x'  —  P'  cos  I  sin  S  cos  (k)  •      S0  =  d  +  x  —  x' ; 
A  a  in  time  =  [8.82391]— ^-  ;  a0  =  a 


8.  In  the  hourly  ephemeris  of  the  moon  fix  on  a  convenient  time  (t)  at 
which  the  moon's  right  ascension  is  near  to  a0,  and  for  this  time  take  out 


TERRESTRIAL   LONGITUDE.  221 

the  right  ascension  (A),  the  declination  (Z>),  and  their  hourly  variations 
-4,,  DI.     Then, 

m  =  ^S—LJ.  ;  to  =  (t)  +  [3.55630]  m  ; 


n  =  [1.17609]  A!  cos  (D)  ; 

cos  ±  =  [0.56463]  ^^. 

p 

Corresponding  Greenwich  mean  time  =  t0  +  [2.99167]  —  sin  (r\  =p  4,). 


Practical  Rules  for  Calculating  the  Longitude  from  an  Observed 
Occultation. 

With  the  estimated  longitude  find  the  corresponding  Greenwich  time 
loughly  to  the  minute,  and  for  this  time  take  out  from  the  ephemeris  the 
moon's  declination  roughly  to  the  minute,  her  horizontal  parallax  to  the 
tenth  of  a  second,  and  the  sun's  right  ascension  in  time  to  the  nearest  sec- 
ond. To  the  sun's  right  ascension  add  the  apparent  time  of  the  observa- 
tion, which  will  give  the  right  ascension  of  the  meridian.  The  difference 
between  this  right  ascension  and  that  of  the  star  will  give  the  hour  angle 
of  the  star  in  time,  which  must  be  reduced  into  arc  in  the  usual  manner  • 
it  will  be 

'  >  when  R.  A.  of  meridian  is  <  f  >•  than  R.  A.  of  *. 

E.    )  (  less        j 

Reduce  the  latitude  of  the  place  by  subtracting  the  correction  found  in 
the  table  in  Appendix  XL,  p.  336,  for  which  the  nearest  correction  found 
in  the  table  will  be  sufficient. 

To  the  proportional  logarithm  of  the  moon's  horizontal  parallax,  add  the 
correction  answering  to  the  latitude  in  the  following  series: 

Lat.      .      0     11     19     24     29     34     38     42     46     50     54     59     64     69     77     90 
Corr.    .         0       1       2       3       4       5       6       7       8       9      10     11     12     13     14 

To  the  proportional  logarithm  of  the  horizontal  parallax,  so  corrected, 
add  the  log.  secant  of  the  reduced  latitude  and  the  log.  cosecant  of  the 
hour  angle.  To  the* sum  (£,)  add  the  log.  cosine  of  the  moon's  declination 
and  the  constant  log.  0.3010.  The  result  will  be  the  prop.  log.  of  an  arc, 
which,  subtracted  from  the  hour  angle,  will  give  the  hour  angle  coirected. 

To  the  corrected  prop.  log.  of  the  horizontal  parallax,  add  the  log.  secant 


222  SPHERICAL    ASTRONOMY. 

of  the  *'s  leclination,  and  the  log.  cosecant  of  the  reduced  latitude.  To 
the  same  log.  add  the  log.  cosecant  of  the  *'s  declination,  the  log.  secant 
of  the  reduced  latitude,  and  the  log.  secant  of  the  hour  angle  corrected. 
These  sums  will  be  the  prop.  logs,  of  two  arcs. 

The  former  arc  to  have  the  same  name  as  the  latitude. 

The  latter  to  have 


a  different  name  from 
the  same  name  as 


!(  less        ) 
the  dec.  when  the  h.  an^le  is  \  v  than  90C 

{  greater  j 


The  sum  of  these  two  arcs,  having  regard  to  their  names,  will  give  the 
correction  to  be  applied  to  the  *'s  declination  to  get  the  declination 
corrected. 

To  the  sum  (S{)  add  the  constant  log.  1.1761,  and  the  log.  cosine  of 
the  *  's  declination  corrected ;  the  sum  will  be  the  prop.  log.  of  an  arc  in 
time,  to  be 

added  to 

subtracted  from 


!the  *'s  R.  A.,  when  it  is  x          [•  of  the  meridian, 
(east  ) 


to  get  the  *  's  right  ascension  corrected. 

In  the  hourly  ephemeris  of  the  moon,  fix  on  a  convenient  time  at 
which  her  right  ascension  is  near  to  that  of  the  star  corrected  ;  and, 
for  this  time,  take  out  the  right  ascension,  the  declination,  and  their  hourly 
variations. 

Subtract  the  common  log.  of  the  difference  between  the  corrected  right 
ascension  of  the  star  and  the  right  ascension  of  the  moon,  from  the  com- 
mon log.  of  the  hourly  motion  in  right  ascension ;  to  the  remainder  add 
the  constant  log.  0.4771 ;  to  the  same  remainder  add  the  prop.  log.  of  the 
hourly  motion  in  declination.  The  former  sum  will  be  the  prop.  log.  of  a 
time  to  be 

added  to  i    .  .    ( greater )    , 

.,  ,        >  the  assumed  time  when  %.  s  R.  A.  is  <f  v  than  D  s  R.  A. 

subtracted  from  \  f  less        ) 

to  get  the  time  corrected. 

The  latter  will  be  the  prop.  log.  of  a  correction  of  the  D 's  declinatior, 
to  be  applied  with 

the  same  name  as         )  ,  .     ( greater )    , 

,.„,  ..         >  hourly  var.  when  %  s  R.  A.  is  <°  V  than  ])  s  R.  A, 

a  different  name  from  \  }  less        ) 

To  the  common  log.  of  the  hourly  motion  in  right  ascension,  add  the 
log.  cosine  of  the  D 's  corrected  declination ;  and  to  the  sum  (S2)  add  the 
pi-op,  log.  of  the  hourly  motion  in  declination  and  the  constant  log.  7.1427. 


TERRESTRIAL   LONGITUDE.  223 

The  result  will  be  the  log.  cotangent  of  the  first  orbital  inclination,*  and 
must  take 


the  same  name  as 
a  different  name  from 


(hourly  motion  in  dec.  when  *  is  •!         ,    I  of 
(  south  j 


To  the  prop.  log.  of  the  difference  between  the  star's  declination  cor-  . 
rected  and  the  moon's  declination  corrected,  add  the  constant  log.  9.43"54t 
and  the  log.  secant  of  the  preceding  orbital  inclination ;  and  from  the  sum 
deduct  the  prop.  log.  of  the  horizontal  parallax.  The  remainder  will  be 
the  log.  secant  of  the  second  orbital  inclination,!  which  must  have  the 
name 


S.1 
N. 


...  ,  immersion 

when  the  observation  is  an 

'  emersion. 


Add  together  the  two  orbital  inclinations,  having  proper  regard  to  their 
names ;  and  to  the  log.  cosecant  of  this  sum  add  the  preceding  sum  (/Sr2), 
the  prop.  log.  of  the  horizontal  parallax,  and  the  constant  log.  8.1844. 
The  sum  will  be  the  prop.  log.  of  a  correction  to  be  applied  to  the  time 
corrected  to  get  the  mean  time  at  Greenwich :  it  must  be 

added         )  (  N. 

,  >  when  the  sum  of  the  orbital  inclinations  is  <  „ 
subtracted  )  (  S. 

By  applying  the  equation  of  time  from  p.  II.  of  the  ephemeris,  there 
will  result  the  Greenwich  apparent  time,  and  the  difference  between  it  and 
the  apparent  time  of  observation  will  show  the  longitude  of  the  place  from 
Greenwich ;  it  will  be 


W. 
E. 


J  t  \  i  cfreater  ) 

,,  >  when  the  Greenwich  time  is  V  f  J-  than  the  observed. 

i.  (  j  less        j 


Examples. 

I.    SOCAR    ECLIPSE. 

For  a  solar  eclipse,  take  the  example  directly  calculated  in  Appendix  XL, 
page  412: 

Suppose  the  beginning  of  the  solar  eclipse  on  May  15,  1836,  to  be  observed  to 
take  place  at  ih  36m  35» -6  p.  M.,  apparent  time,  in  latitude  55°  67'  20"  N.,  and 
longitude  about  i  am  "W. 


*  With  the  parallel  of  declination.  f  With  the  moon's  limb. 


224  SPHERICAL  ASTROtfOM* 

Here  we  have 

h.  m. 

Observed  apparent  time     .      i  36-6 

Longitude 12-0  A  =  +ih  56m  35S«6 

Greenwich  apparent  time         i     48-6  =+        24°     8' -9 

Equation  of  time      ...  3  •  9 

Greenwich  mean  time    .      .      i     44*7 

We   hence    take    from    the    ephemeris,    a  =  3h    29™     19",    <5  =  +i8°    67' -t 
ff  =  i5'  49"-9,  -#  =  +19°  19',  P  =  54'  24'' -4,  7r  =  8".5,  P  — 77  =  54'  i5"«9. 

Latitude  +  55°  57'  20" 
Reduction  10    28 


.     +  55    46  52     .      .      .      .     0  =  9-99902 

p  —  573.51267  cos  (h)  +9  -96060       .     +9-96060 

p     .     9.99902  cot/     4-9-83256  cos  I  +  9-  ySooi 

P'  .     3.51169  0-f3i   5o-7  tan  6    -{-9.79316  G       +9-71061 
cos^    9-7500T 


sinA+9-6n83  0+<5+5o  48-3  cos       +9-80069  B  +9-78899 

p       +2-87353  (i)  +9-92162  check +9 -921 6-J 

cos  D  9-97484  tan  (7<)+9- 64936 

+  2-89869  tan  M  +9- 57098 

k+°4     8.0  const.   7-92082  cos  M  +9T^       .  +9.97,80 

A&  +          6-6            +0-81951  tan  (0+(5)+o- 08861    cose  +9-81719 

(h)  +24     2-3                                  £+48°  58'. 3  tan  £    +0-06041  B  +9-78899 

*+Iw  p>  +L!i!5 


+  19  3o-9 

cos       9 

•9743o  (2) 

a      .       i5' 

49' 

'•9 



+  3 

-89923  (i)-(2) 

A(T 

1  1 

-6  P     . 

3-5i38o 

const.  8 

-82391 

<r0     .       15 

38 

-  const. 

9.43537 

(log. 

-     +i 

.723i4 

«     .     14 

49 

-6 

2-94917 

1   A«- 

f  Oh     om 

52^.86 

A     .      3o 

27 

•9 

a 

3   29 

J9 

ao      3     30       12 

By  inspecting  the  hourly  ephemeris  of  the  moon's  right  ascension  on  May  15th 
with  a0  =  3h  3om  1 2s,  the  most  eligible  time  to  assume  is  evidently  (£)  =  3h  om  os ;  at 
this  time  we  have  (^4)=  3h  3om  42s- 84,  (AJ  =2ra  os-68,  (D)  =  +  19°  3i'  34"-o, 
(A)  =  +9'  55"- 2,  (a)  =  3h  29-1  3is.57,  (ai)  =  +  9'-89,  (J)  =  +i8°  58'  21". 4, 
(^i)  =  +  34 "*8 :  with  these  we  proceed  as  follows  : 

(AJ     .      .    "a     0-68  (A)   .      -     +  9  55-2 

(«i)       -      -  9'89  (<M     •      -     +      34-8 

At       .      .     i  5o-79  Di      .     .     +  9  20.4 


log 


TERRESTRIAL    LONGITUDE. 


225 


A  a       -f 

h.    m.    s. 
3  29  3i-57 
o  52-86 

w 

O        '          " 

+     18  58  21-4 
+          33  18.4 

(«o)        • 

3  3o  24-43 

Co) 

-f     19  3i  39.8 

(A)        . 

3  3o  42-84 

-(A)      . 

—  o  18-41 

A, 

—  i-265o5 
2-o445o 

D,  . 

.     +2.7485o(i) 

in 

—  9  •  22o55 

—  9«22o55 

const. 

3-5563o 

(log- 

-     —1-96905 

j  log-      - 

—  2.77685 

Oh    Qm    58s.2 

\ 

V>) 

—    o°  i'  33".  i 
+  19  3i    34  .0 

(t)  3     o       o 

t 


3o 


-f  2  5o      i  -8 


19  3i    39  .8 


38 


•9 


cos 


const. 


9.97428 


i > i 7609 


n   . 
o 

v    .      .     —  19  4i  •  2               tan  n   . 

COS  rj     . 

k   . 
A     . 

<//    .      .     +  92  55-2              cos  \\>  . 

9  —  ;p    .      .    —  112  36-4          •    sin 
A    . 
const.    . 

3.t9487(2) 
.     —^5363  (i)  — 

.     +  9-97384 
.     —  i  -99520 

—  i  -  96904 
3.26196 

.     —8-70708 

.     —9.96528 
3-26196 
3-55630 

—  6- 78354  (3) 
corr.  —  r»  4ra  38s- 5 —  3-58867  (3)  — (2) 

<0-f-corr.  +   i  45     23-3     Greenwich  mean  time. 
3     56  «o    Equation  of  time. 


i   49     19  -3     Greenwich  apparent  time, 
i  36    35  -6    Observed  "          " 


Longitude 


12     43  >  7    W.  of  Greenwich. 


15 


226 


SPHERICAL    ASTRONOMY. 


II.    OCCTJLTATION    OF   A    STAR. 


Suppose,  at  Bedford,  on  January  7,  1836,  in  latitude  52°  8'  28"  N.,  the  immer- 
sion of  i  Leonis  to  be  observed  at  ioh  39m  22S'4  P.  M.,  apparent  time,  and  tho 
estimated  longitude  to  be  about  oh  im  "W.  Required  the  longitude?1 


Apparent  time  (observation) 
Longitude 


h.    m. 
10  39 
o     t  W. 


Latitude 
Reduc. 


O 

N.  62 


8  28 
10  5 


Apparent  time  (Greenwich 
Equation  of  time 

Mean  time  (Greenwich) 


10 


K  5i  57  3r 
Reduced  or  geocentric  latitude. 


10  4? 


For  Jan.  7,  at  ioh  4jm,  we  find,  from  the  Ephemeris,  ©'s  R.  A.  =  19''  I2m  4o8 
D's  dec.  =  N.  i5°  5o',  and  ]> 's  equ.  hor.  par.  =56'  i"-$. 


h.    m.    s. 

0's  R.  A. 

19  12  4o 

P.  L.   D  's  hor.  par. 

o-5o68 

App.  tim< 

3    

IO    39    22 

corr.  for  lat.     . 

9 

R.  A.  meridian        .... 

5    52       2 

P.  L.  corr'1.  hor.  par    . 

0-5077 

«         .j 

C            •       •       •       •       • 

IO    23    26 

sec.  red.  lat. 

o  •  2  io3 

n 
s|c's  hour 

iin  time  . 
in  arc     . 

4  3i  24 
67°  5i' 

cosec.  hour  angle 
sum  (Si)    .... 

o-o333 

cos.  J)  's  dec.   . 

9-9832 

const,  log. 

o-3oro 

corr".    .      . 

17     .     . 

P.  L.  corr".      .      .      . 

7^355 

sjc's  hour 

angle  E.  corrd.    .      . 

67  34 

P.  L.  cortd.  hor.  par. 

' 

o-5o77 

sec.  j|c's  dec. 

.     O'OiSo 

0-5876 

cosec.  red.  lat. 

o«io37 

sec. 

o  •  2io3 

O       '        " 

N.    o  42  33-o 

P.  L.  0-6264 

sec.  corrd.  hour  angle 

o-4i84 

S.     o    3  23-9 



P.  L  

1-7240 

corr". 

.     N.    o  39    9-1 

sum  (Si)    .... 

o.75r3 

ilc's  dec 

K  i4  58  38-8 

5Jc's  dec.  c 

• 

cos 

o-o836 

iorr''.  N.   1  5  37  47*9 

corr".    . 

O1'     2m    1  2s  -56 

P.  L.  corr".     . 

y  yu--"j 

1.9110 

#'s  R.  A.  . 

.     10  23     26  .39  • 

*'s  R.  A.  corrd.  10  2t     i3  -83 

On  referring  with  the  >}c's  corrected  R.  A.  to  the  hourly  ephemens  of  the  moon, 
it  will  evidently  be  most  convenient  to  take  out  the  data  at  nh;  for  this  time  \ve 
have  D  's  R.  A.  —  lo'1  2om  58* -47,  hourly  motion  D  's  R.  A.  —  2™  2" -9,  ])  '*  dec.  = 
N.  1 5°  47'  u"-o,  hourly  motion  D's  dec  =S.  »i'  4i"-5. 


TERRESTRIAL    LONGITUDE  227 


h.    in.    s. 

ijc'scorrd.  R.  A.      10  21   i3-83 
P'sR.  A.     ,     .     10  20  58-4? 


diff.      .     .  o  i5*36 


(  common  log       .      .      i-i864 
com,  log.  h.  m.  }  's  R.  A.  2-0896 

Remainder     .     0.9082    ,.,....,,...     o«oo32 
const.log..      .      .      .     0.4771         P.  L.  h.  m.  1> 's  dec.       .      .      .      1-1874 


h.   in.   8. 


o 


corr".        .      .     o  7  29-9  RL.  i-38o3       corr".       .     S.     o     i    27-7  P.  L.  2-090*: 
Time  assumed  1 1  o     o  D  's  dec.       .     K".  1 5  47  1 1  •  o 

Time  corra.    .   u  7  29-9  J> 's  dec.  corr'',  K  i5  45  43-3 


com.  log.  h.  m.  D  's  R.  A. 

cos.  D  's  corrd.  dec. 

.     2-0896 
.     9'9834 

jfc's  corrd.  dec.  . 
D's     "         "     . 

O        ' 

.     N.  »  5  37  47^9 
.     N.  i5  45  43-3 

sum  (S»)                 .      »      • 

2-0730 

<diff  (#  S  of  ' 

b  }                  ~7  55-  J 

"  /                   ' 

P.  L.  h.  m.  &  's  dec.     .      . 
const,  lo0*         .... 

.     1-1874 
7*1427 

<P.L.       .      . 
const  log 

.     .     1-3563 
0.4354 

1st  Orb.  incL    N.  21°  34' 

cot.  o«4o3i 

P.  L.  D  's  hor.  par.   .     o  -  5o68 
2nd  Orb.  incl.    S.  61      9 sec o-3i64 

»um  ,     S.  89    35       .....    cosec 0-1957 

sum  (&)        .      .      .     2-0730 
P.  L.  ]>  's  hor.  par.  .     o  -  5o68 

const,  log.     .      .      .     $.i844 
h.    m.    is. 

corr".       o  19  44-5         P.  L.  .      .     ,     »     „     ©.9599 
Time  corrd.     n     7  29-9 

Greenwich  mean  time     10  47  45-4 
Equation  of  time  .      .  6  3 1 » o 

Greenwich  app.  time        10  4i    i4'^ 
Observed       *       "  10  39  22>4 

Longitude        .     .     ,  t  5a-o  "W 

P.  S. — The  principle  of  reversing  the  effect  of  the  relative  horizontal  parallax 
on  the  position  of  the  sun,  instead  of  using  the  actual  effect  on  the  position  of  the 
moon,  may  be  advantageously  employed  in  the  direct  calculation  of  an  eclipse  fof 
a  particular  place.  It  will  only  be  necessary  to  use  the  parallaxes  for  the  sun 
viewed  as  an  apparent  position,  and  to  diminish  the  semi-diarneter  by  the  amount 
derived  from  the  table  on  page  360.  Thu?,  it  appears,  at  the  beginning  of  the 
eclipse,  for  instance,  that  the  contact  may  be  mathematically  tested  in  two  ways. 
First,  we  may  apply  the  actual  effects  of  the  parallax  to  the  true  position  of  the 
moon,  then  augment  her  semi-diameter,  and  thus  establish  ft  contact  of  the  limbs. 
But,  if  we  reverse  the  operation,  and  consider  the  sun  to  be  an  apparent  body 
under  the  influence  of  the  relative  parallax^  then  clearing  it  from  this  supposed 


228  SPHERICAL    ASTRONOMY. 

influence  by  reversing  the  parallax,  and  diminishing  the  semi-diameter,  a  contact 
will  similarly  be  established  with  the  true  limb  of  the  moon ;  and  this  principle, 
in  its  application  to  solar  eclipses,  possesses  an  advantage  similar  to  that  derived 
in  the  case  of  an  occultation,  by  considering  the  star  as  an  apparent  place.  (See 
Appendix  XL,  page  899  )* 

The  formulae,  Nos.  2,  3,  4,  and  5,  pp.  406,  40Y,  may,  according  to  thi& 
method,  be  supplied  by  the  following : 

2.         P'  =  P(P-«r);  m  =  P'cosl; 

$,  =  [9.4180]  ;  Q2  =  [9.4180}  m  sin  d ; 

s=  [9.43537]  -P. 


cos  JJ 
A  A  in  minutes  =  [7.92082]  k  sin  h ; 

tan  6  =  cos  (A)  cot  l\  Gf  =  cos  (h)  co»  t, , 

tan  M= -r. FX  tan  (A) ;  tan  s  =  tan  (&  +  8)  cos  M\ 

cos  (6  -f-  o) 


Check      .       . 


=  cos  M  cos  s  ; 
sind 


.  — 

cos   &+o         B' 


(fQ=.(f  —  diminution  for  s  • 

(  partial  )  .       j  s  -f  <f 

For  <  ^  .      >  phase,  Z.  =  1 

(  total  or  annular  )  (  5  ^*  tf0. 


A  at  =  ^!  A?0  cos  h  ;  A  5,  =  Q.2  sin  (A). 

o  =  5  -f-  A  <5  ;  a'  =  a  —  A  a  ; 

y  =  (a  —  A  a)  cos'Z>  ;  3^  =  (at  —  A  a,)  cos  D  ; 

a:  =?  (Z>  +  a'  corr.)  —  <50  ;  ^  =  Dl  —  A  ^. 


*  This  was  inadvertently  ascribed  to  Carlini.  Professor  Henderson,  by  whom 
a  paper  has  appeared  upon  this  very  point  in  the  Quarterly  Journal  for  1828, 
page  411,  informs  me  that,  the  method  has  been  long  in  practice,  and  that  it  was 
employed  at  an  early  period  by  Dr,  Maskelyne. 


CALENDAR.  229 

§  734.  Longitude  ly  Eclipses  of  Jupiter's  Satellites. — The  eclipses  of 
Jupiter's  satellites  are  computed  in  advance,  and  the  times  of  occurrence 
inserted  in  the  Nautical  Almanac,  to  facilitate  the  determination  of  terres- 
trial longitude.  After  ascertaining,  by  inspection,  about  the  time  an 
eclipse  begins  and  ends,  the  satellites  are  watched  with  a  good  telescope, 
and  the  precise  local  time  of  entrance  into  and  departure  from  the  shadow 
noted  as  nearly  as  possible.  The  time  given  in  the  Almanac,  diminished 
by  this  observed  local  time,  is  the  longitude ;  west,  when  the  difference  is 
positive,  east  when  negative.  This  method  for  finding  longitude  is  defec- 
tive, for  reasons  stated  in  §  497. 

CALENDAR. 

§  735.  To  divide  and  measure  time  and  to  note  the  occurrence  of 
events  in  a  way  to  give  a  distinct  idea  of  their  order  of  succession  and 
the  intervals  of  time  between  them,  is  the  purpose  of  Chronology. 

§  736.  All  measurements  require  standard  units.  These  units  are,  for 
the  most  part,  purely  arbitrary,  and  are  equally  convenient  in  practice. 
But  such  is  not  the  case  in  chronology.  Time  is  divided  and  marked  by 
phenomena  which  are  beyond  our  control,  and  which  indeed  regulate  our 
wants  and  occupations.  The  alternation  of  day  and  night  forces  upon  us 
the  solar  day  as  a  natural  unit  of  time. 

§  737.  To  avoid  the  use  of  numerous  figures  in  the  expression  of  great 
magnitudes,  all  measurements  must  have  their  scales  of  large  and  small 
units,  and  usually  the  selection  of  the  larger  is  as  arbitrary  as  the  smaller  ; 
Out  here  the  phenomena  of  nature  again  interpose,  and  the  periodical 
return  of  the  seasons,  upon  which  all  the  more  important  arrangements 
and  business  transactions  of  life  depend,  prescribes  the  tropical  year  as  an- 
other and  higher  order  of  unit  in  chronology. 

§  738.  But  the  solar  day  and  tropical  year  are  both  variable,  and  are 
therefore  wanting  in  all  the  essential  qualities  of  standards.  Neither  are 
they  commensurable  the  one  with  the  other ;  they  are  on  this  account 
unfit  units  for  the  same  scale.  .Iu  the  measurement  of  space,  for  instance, 
each  unit  is  constant,  and  one  is  an  aliquot  part  of  another — a  yard  is 
equivalent  to  three  feet,  a  foot  to  twelve  inches,  <fec.  But  a  year  is  no 
exact  number  of  days,  nor  an  integer  number  and  any  exact  fraction,  as  a 
third  or  a  fourth,  even ;  but  the  surplus  is  an  incommensurable  fraction 
which  possesses  the  same  kind  of  inconvenience  in  the  reckoning  of  time 
that  would  arise  in  that  of  money  with  gold  coins  of  101  dimes  and  odd 
«ents,  and  a  fraction  over.  For  this  there  would  be  no  remedy  but  to 


230  SPHERICAL    ASTRONOMY. 

keep  an  accurate  register  of  the  surplus  fractions,  and  when  they  amount 
to  A  whole  unit,  to  cast  them  over  to  the  integer  account.  To  do  this  in 
the  simplest  and  most  convenient  manner  in  the  reckoning  of  time,  is  the 
object  of  the  calendar. 

§  739.  A  calendar  is,  therefore,  a  classification  of  the  natural  and  other 
divisions  of  time,  with  such  rules  for  their  application  to  chronology  as 
shall  take  into  account  every  portion  of  duration  without  recording  any 
one  portion  twice. 

These  divisions  are  years,  months^  weeks,  days,  and  certain  periods,  to 
be  noticed  presently,  and  which  are  chiefly  important  in  the  use  made  of 
them  in  fixing  upon  a  common  epoch  or  origin  of  reference. 

§  740.  Julian  Calendar. — The  years  are  denominated  as  years  current, 
not  as  years  past,  from  the  midnight  between  the  31st  of  December  and 
1st  of  January,  immediately  subseqiaent  to  the  birth  of  Christ,  according 
to  the  chronological  determination  of  that  event,  and  this  origin  is  desig- 
nated by  the  letters  A.  D.  or  B.  C.,  according  as  the  year  is  subsequent  ov 
previous.  Every  year  whose  number  is  not  divisible  by  four  without  a  re- 
mainder, consists  of  365  days,  and  every  year  which  is  so  divisible  of  366. 
The  additional  day  in  every  fourth  year  is  called  the  Intercalary  day. 
The  years  which  consist  of  36-5  days  are  called  Common  years  ;  those  which 
consist  of  366  days  are  called  Bissextile  years,  and  frequently  Leap  years. 
The  mean  length  of  the  year  by  this  rule  is  obviously  365J  days,  and  the 
mode  of  reckoning  time  by  this  unit  in  the-  way  just  described  is  called  the 
Julian  Calendar. 

§  741.  The  year  is  divided  into  12  months  of  unequal  length.  They 
are  named,  in  order  of  succession,  January,  February,  March,,  April,  May, 
June,  July,  August,  September,  October,.  November,  and  December. 
January,  March,  May,  July,  August,  October,  and  December,  have  each 
31  days  ;  each  of  the  others  except  February  has  30,  and  February  has  in 
a  common  year  28  and  in  a  bissextile  year  29 ;  so  that  the  intercalary  day 
is  added  to  February.  The  weeks  consist  of  seven  days,  named  in  order, 
Sunday,  Monday,  Tuesday,  Wednesday,  Thursday,  Friday,  and  Saturday. 

§  7  12.  Gregorian  Calendar. — The  Julian  year  consists,  of  365.25  days ; 
the  tropical  year  of  365.24224,  making  the  Julian  longer  than  the  tropical 
by  0.00776  of  a  day,  and  causing  the  seasons  to  begin  earlier  and  earlier 
every  year  as  designated  by  the  Julian  dates.  In  process  of  time  the 
seasons  would  therefore  correspond  to  opposite  dates  of  the  yearr  and  as 
this  was  likely  to  interfere  with  the  times  of  holding  certain  church  fes- 
tivals, Pope  Gregory  XIII.  determined  upon  a  reformation  of  the  Julian 
calendar. 


CALENDAR.  231 

§  743.  In  A.  D.  325,  the  seasons,  festivals,  and  Julian  dates  corres- 
ponded  with  one  another,  according  to  church  rule.  The  reformation  was 
effected  in  1582.  Now  (1582  —  325)  X  Od.00776  =  9d.6243.  Again, 
0>!.00776  X  400  =  3d.104.  The  Pope  ordered  that  the  day  following,  the 
4th  of  October,  1582,  should  be  called  the  15th  instead  of  the  5th.  This 
brought  the  date  of  the  sun's  entering  the  vernal  equinox  to  what  it  was 
in  325,  the  time  of  holding  the  Council  of  Nice.  And  to  secure  this 
coincidence  in  future,  he  also  ordered  that  three  intercalary  days  should  be 
omitted  every  four  hundred  years,  the  omissions  to  take  place  in  those 
centennial  years  which  are  not  divisible  by  400  ;  so  that  1700,  1800,  and 
1 900,  which  by  the  Julian  mode  of  reckoning  are  bissextile,  are  made  b1* 
the  Gregorian  common  years.  There  is,  therefore,  at  the  present  time, 
viz.,  in  the  19th  century,  a  difference  of  12  days  between  the  Julian  and 
Gregorian  dates.  The  mode  of  reckoning  by  the  Julian  calendar  is  called 
Old,  and  that  by  the  Gregorian  New  Style.  New  style  is  followed 
throughout  Christendom  except  in  Russia,  where  the  old  style  is  pre- 
served. 

§  744.  Solar  Cycle. — This  is  a  period  of  28  Julian  years,  after  the  lapse 
of  which  the  same  days  of  the  week  in  the  Julian  system  would  return  to 
the  same  days  of  each  month  throughout  the  year.  For  four  such  years 
consist  of  1461  days,  which  is  not  a  multiple  of  7,  but  7  times  4  or  28 
years  is  a  multiple  of  7.  The  place  in  this  cycle  for  any  year  of  A.  D.  is 
found  by  adding  9  to  the  year,  dividing  by  28,  and  taking  the  remainder. 
When  there  is  no  remainder,  the  number  sought  is  28. 

§  7^5.  Lunar  Cycle. — This  is  a  period  of  19  years  or  235  lunations  which 
differ  from  19  Julian  years  only  by  about  an  hour  and  a  half;  so  that, 
supposing  the  new  moon  to  happen  on  the  first  of  January  in  the  first  year 
of  the  lunar  cycle,  it  will  happen  on  that  day  or  within  a  very  short  time 
of  its  beginning  or  ending  again  after  the  lapse  of  19  years.  The  number 
of  the  year  of  the  lunar  cycle  is  called  the  golden  number,  to  find  which 
•tdd  1  to  the  number  of  the  year  A.D.,  and  take  the  remainder  after 
dividing  by  19.  If  there  be  no  remainder,  the  golden  number  will  be  19. 
The  golden  number  is  used  in  ecclesiastical  dates  to  determine  the  civil 
date  of  Easter. 

§  746.  Cycle  of  Indiction. — This  is  a  period  of  15  years,  used  in  the 
courts  of  law  and  in  the  fiscal  organization  of  the  Roman  empire,  and 
thence  introduced  into  legal  dates  as  the  golden  namber  into  the  ecclesias- 
tical. To  find  the  place  of  any  year  of  A.  D.  in  the  cycle  of  indictkm,  add 
3,  divide  by  15,  and  take  the  remainder.  If  there  be  no  remainder,  the 
number  sought  will  be  15. 


232  SPHERICAL     ASTRONOMY.      • 

§  747.  Julian  Period.— The  product  of  28,  19,  and  15  is  7980.  This 
is  called  the  Julian  Period  ;  and  it  is  obvious  that  after  this  period,  the 
years  of  the  solar,  lunar,  and  indlctloi.  cvciea  will  recur  in  the  same  order ; 
that  is,  each  year  wil»  holci  the  same  place  in  all  the  three  cycles  as  the 
corresponding  year  in  the  previous  period. 

§  748.  As  no  common  factor  exists  in  the  numbers  28,  19,  and  15,  it  is 
plain  that  no  two  years  in  the  Julian  period  can  agree  in  its  three  compo- 
nent cycles,  and  to  specify  the  number  of  a  year  in  each  of  the  latter  is  to 
specify  the  number  of  the  year  in  the  Julian  period,  which  now  embraces 
the  entire  authentic  chronology.  The  first  year  of  the  current  Julian  period, 
or  that  of  which  the  number  of  the  three  subordinate  periods  is  1,  was  the 
year  B.  C.  4713,  and  noon  of  the  1st  of  January  of  that  year,  for  the  me- 
ridian of  Alexandria  in  Egypt  is  the  chronological  epoch  to  which  all  his- 
torical eras  are  most  readily  referred,  by  computing  the  number  of  integer 
days  intervening  between  it  and  Alexandria  noon  of  the  days  which  serve 
as  the  respective  epochs  of  these  eras.  The  meridian  of  Alexandria  is 
chosen,  because  it  is  that  to  which  Ptolemy  refers  the  commencement  of  the 
era  of  Nabonassar,  the  basis  of  all  his  calculations. 

§  749.  Given  the  year  of  the  Julian  period,  those  of  the  subordinate 
cycles  are  found  as  above.  Conversely,  given  the  year  of  the  solar,  lunar, 
and  indiction  cycles,  to  determine  the  year  of  the  Julian  period,  proceed  as 
follows,  viz. :  Multiply  the  number  of  the  year  in  the  solar  cycle  by  4845, 
in  the  lunar  by  4200,  and  in  the  indiction  by  6916,  and  divide  the  sum  of 
the  products  by  7980,  and  the  remainder  will  be  the  year  of  the  Julian 
period  sought. 

§  750.  A  date,  whether  of  a  day  or  year,  always  expresses,  as  before  re- 
marked, the  day  or  year  current,  not  elapsed  ;  and  the  designation  of  a  year 
by  A.  D.  or  B.  C.  is  to  be  regarded  as  the  name  of  that  year,  and  not  as  a 
mere  number  designating  the  place  of  the  year  in  a  scale  of  time.  Thus, 
in  the  date  January  5,  B.  C.  1,  January  5th  does  not  mean  that  5  days  in 
January  have  elapsed,  but  that  4  have  elapsed,  and  the  5th  is  current. 
And  B.  C.  1,  indicates  that  the  first  day  of  the  year  so  named  (the  first, 
current  before  Christ)  preceded  the  first  day  of  the  common  era  by  one 
year.  The  scale  A.  D.  and  B.  C.  is  not  continuous  ;  the-year  0,  is  wanting 
in  both  parts,  so  that  supposing  the  common  reckoning  correct,  our  Saviour 
was  born  in  the  year  B.  C.  1. 

§  751.  Epact. — The  mean  age  of  the  moon  at  the  commencement  of  a 
year  is  called  the  epact.  It  is  a  name  given  to  the  interval  of  time  be- 
tween the  first  of  the  year  and  the  next  preced  ng  mean  new  moon:  it  is 
expressed  in  days,  hours,  minutes,  and  seconds.  Its  use  is  to  find  the  days 


CALENDAR.  233 

of  mean  new  and  full  moon  throughout  the  year,  and  thence  the  dates  of 
certain  church  festivals. 

§  752.  Equinoctial  Time. — Astronomical  time  reckons  from  noon  of 
the  current  day ;  civil,  from  the  preceding  midnight.  Astronomical  and  civil 
dates  coincide,  therefore,  only  during  the  first  half  of  the  astronomical  and 
last  half  of  the  civil  day.  Were  this  the  only  cause  of  discrepancy,  it  might 
be  remedied  by  shifting  the  astronomical  epoch  to  coincide  with  the  civil. 
But  there  is  an  inconvenience  to  which  both  are  liable,  inherent  in  the 
nature  of  the  day  itself,  which  is  a  local  phenomenon,  and  commences  at 
different  instants  of  absolute  time  under  different  meridians.  In  conse- 
quence, all  astronomical  observations  require  to  be  given,  to  render  them 
comparable  with  one  another,  in  addition  to  their  date,  the  longitude  of 
the  place  of  observation  from  some  known  meridian.  But  even  this  does 
not  meet  the  whole  difficulty,  for  when  it  is  Monday,  1st  of  January,  of  any 
year,  in  one  part  of  the  world,  it  will  be  Sunday,  31st  December,  of  the 
preceding  year,  in  another  part  of  the  world,  so  long  as  time  is  reckoned 
by  local  hours. 

The  equivoque  can  only  be  avoided  by  reckoning  time  from  an  epoch 
common  to  all  the  earth.  Such  an  epoch  is  that  which  marks  the  passage 
of  an  imaginary  sun  having  a  mean  motion  equal  to  that  of  the  true  sun, 
through  a  mean  vernal  equinox  receding  uniformly  upon  the  ecliptic  with 
a  motion  equal  to  the  mean  motion  of  the  true  equinox.  Time  reckoned 
from  this  epoch  is  called  equinoctial  time.  Equinoctial  time  is  therefore 
the  mean  longitude  of  the  sun  converted  into  time  at  the  rate  of  360°  to 
the  tropical  year. 


APPENDIX. 


236 


APPENDIX    I. 


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APPENDIX   II.  237 


APPEI^  DIX   II. 

ASTRONOMICAL  INSTRUMENTS. 

Astronomical   Clock  and   Chronometer, 

1.  —  The  order  and  succession  of  celestial  phenomena  make  time  a 
most  important  element  in  astronomy,  and  accordingly  the  utmost  scientific 
and  mechanical  skill  has  been  devoted  to  the  perfection  of  instruments 
to  indicate  and  measure  its  lapse ;  §  37. 

The  best  time-keepers  now  in  use  are  the  Clock  and  Chronometer.  Both 
consist  essentially  of  a  motor,  a  combination  of  wheel-work  to  transmit  and 
qualify  the  motion  it  impresses,  and  a  check,  alternately  to  arrest  and 
liberate  the  movement,  and  thus  to  mark  an  interval  designed  to  be  some 
aliquot  part  of  a  day,  the  natural  unit  of  duration. 

2.  —  The  Clock.  --Iu  the  clock,  the  motor  is  a  weight  A  suspended 
from  a  cord  wound  about  the  drum  B  of  a  wheel  (7,  and  the  check  is 
the  anchor  escapement  JV,  controlled  by  the  vibrations  of  a  pendulum  P, 
whose  rod  is  geared  to  an  arm  projecting  from  the  axis  0,  with  which  the 
anchor  is  firmly  connected.     The  weight  A  turns  the  drum  B  and  its 
wheel  (7;  the  wheel  C  turns  the  pinion  D  and  its  wheel  E\  the  latter 
turns  the  pinion  F  and  its  wheel  6?,  and  so  on  to  the  pinion  L  and  its 
wheel  Mj  called  the  scape-wheel,  of  which  the  teeth  are  considerably  under- 
cut, so  as  to  turn  their  points  in  the  direction  of  the  motion.     The  flukes 
of  the  anchor  are  turned  inward,  forming  two  projections  called  pallets. 
The  distance  between  the  ends  of  the  pallets  is  less  than  that  between  the 
points  of  two  teeth  that  lie  nearest  the  line  drawn  from  one  pallet  to  the 
other ;  and  no  £wo  teeth  can,  therefore,  pass  the  same  pallet  without  the 
wheel  being  arrested  by  the  contact  of  a  tooth  op  the  opposite  side  with 
the  other. 

With  the  swing  of  the  pendulum  the  anchor  oscillates,  and  one  pallet 
is  thus  made  to  approach  while  the  other  recedes  from  the  wheel.  As  soon 
as  the  receding  pallet  disengages  itself  from  a  tooth,  the  wheel  is  turned 


238 


SPHERICAL   ASTIION'OMY 

Fig.  6. 


by  the  motor  and  intermediate  machinery  till  arrested  by  the  approaching 
pallet,  now  interposed  between  its  teeth  on  the  opposite  side.  The  re- 
turning swing  of  the  pendulum  reverses  the  pallet  motion,  liberates  the 
wheel  long  enough  for  another  tooth  to  pass,  and  again  arrests  it,  and 
so  on. 

Thus,  by  regulating  the  length  of  the  pendulum  and  number  of  teeth 
on  the  scape-wheel,  an  index  or  hand  connected  with  the  arbor  of  the 
latter  may  be  made  to  travel  by  successive  leaps,  as  it  were,  around  the 
<'ircu inference  of  a  circle  on  the  dial-plate  in  any  given  time, 

3,  —  If  the  anchor  be  connected  with  the  seconds  pendulum,  and 
there  be  sixty  teeth  on  the  wheel,  each  leap  will  mark  a  second.     The 


APPENDIX   II.  239 

motions  of  the  minute  and  hour  hands  are  regulated  by  suitably  propor- 
tioning the  relative  dimensions  of  the  intermediate  wheels  with  whose 
arbors  these  hands  are  connected. 

4. — The  scape-wheel  being  in  a  state  of  constant  tension  by  the 
incessant  action  of  the  motor,  its  teeth  must  act  upon  the  pallets  first  by 
a  blow  and  then  by  a  pressure  during  the  time  of  contact.  The  bearing 
surfaces,  of  which  there  are  two  on  each  pallet,  inclined  to  one  another, 
are  so  cut  that  the  direction  of  the  blow  on  the  first  from  the  tooth  of 
the  scape-wheel  passes  through  the  axis  of  the  pendulum's  motion,  while 
the  pressure  from  the  same  tooth  on  the  second  passes  clear  of  that  axis 
and  accelerates  the  motion  in  the  direction  of  the  swing,  thus  restoring 
whatever  of  loss  may  come  from  friction  and  atmospheric  resistance. 

5.  —  The  pendulum  bob  possesses  the  principle  of  compensation.    It 
consists  of  a  cylindrical  glass  vessel  resting  upon  a  plate  at  the  end  of  the 
pendulum  rod.     This  vessel  is  filled  with  mercury  to  a  depth  so  adjusted 
to  the  length  of  the  rod  as  to  elevate  by  its  expansion  or  depress  by  its 
contraction  the  centre  of  oscillation  just  as  much  as  this  centre  is  depressed 
by  the  expansion  or  elevated  by  the  contraction  of  the  rod  during  a 
change  of  temperature.     The  distance  between  the  axes  of  suspension  and 
of  oscillation  being  thus  made  invariable,  the  time  of  vibration  will  con- 
tinue constant,  and  the  check  be  interposed  at  equal  intervals. 

6.  —  Chronometer. — The   chronometer   is   an   accurately   constructed 
balance  watch,  uniting  great  portability  with  extreme  accuracy.     It  is  of 
various  sizes,  the  larger  having  dial-plates  from  three  to  four  inches  in 
diameter,   and  running  from  two  to  eight  days  between  the  windings. 
The   larger  kind   are  suspended  upon  gimbals  to  secure   uniformity  of 
position,  are  mounted  in  boxes,  and  are  called  box  chronometers.     The 
smaller  kind  resemble  in  shape  and  size  a  common  watch,  are  worn  in  the 
pocket,  and  are  called  pocket  chronometers. 

1.  —  The  motor  is  an  elastic  spiral  spring  inclosed  in  a  short  cylin- 
drical box  A,  called  the  barrel,  one  end  being  permanently  fastened  to  a 
stationary  axis  E,  about  which  the  barrel  freely  turns,  and  the  other  to 
the  inner  surface  of  the  barrel. 

The  barrel  being  turned  in  the  direction  of  the  coils  of  the  spring,  the 
elastic  force  of  Ae  latter  is  brought  more  and  more  into  play,  and  its 
variable  action  thus  produced  is  communicated  by  means  of  a  chain  B  tc 
a  variable  lever  (7,  called  a,  fusee,  whose  office  is  to  modify  and  transmit  il 
uniformly  to  the  works  of  the  instrument. 

The  fusee  is  a  conical  solid  having  its  surface  broken  into  a  spiral 
shoulder,  running  from  one  end  to  the  other,  the  curve  being  so  regulated 


SPHERICAL   ASTRONOMY 


that  the  distance  of  any  one  of  its  points  from  the  axis  of  the  fusee's  mo- 
tion multiplied  into  the  force  of  the  spring,  acting  through  the  intermedium 
of  the  chain,  shall  be  a  constant  quantity ;  and  as  the  main  wheel  D,  which 


gives  motion  to  the  rest,  is  firmly  secured  to  the  fusee,  the  motion  is  made 
to  act  uniformly  upon  the  instrument. 

8.  —  The  swings  of  the  pendulum  by  which  the  check  was  alternately 
interposed  between   and  withdrawn  from  the  teeth  of  the  scape-wheel  in 
the  clock,  are,  in  the  chronometer,  replaced  by  the  vibrations  of  what  is 
called  the  balance.     This  consists  of  a  wheel  £,  freely  movable  about  an 
axis  (7,  and  a  thin  spiral  spring  S,  one  end  of  which  is  securely  fastened 
to  the  hub  of  the  wheel,  and  the  other  to  a  fixed  support  A.     If  when  the 
spring  is  free  from  tension,  the  wheel  Fig.  8. 

be  brought  to  rest  it  will  remain  so, 
just  as  a  pendulum  bob  brought  to 
rest  at  its  lowest  point  will  remain  im- 
movable. If  from  this  position  of  the 
wheel  it  be  turned  in  either  direction 
about  its  axis,  the  spring  will  wind  or 
unwind,  the  elastic  force  of  the  spring  will  be  called  into  play,  and  will, 
when  the  wheel  is  unobstructed,  carry  it  back  to  its  position  of  equilibrium. 
But  having  reached  this  position,  its  living  force  carries  it  beyond  ;  the 
action  of  the  spring  is  reversed,  and,  after  destroying  the  living  force, 
will  reverse  the  motion;  the  wheel  will  return  to  its  position  of  equilibrium, 
which  it  will  reach  with  a  living  force  equal  to  that  it  had  befo-e  at  the 
same  place,  but  in  a  contrary  direction.  The  wheel  will*pass  on,  the  action 
of  the  spring  be  reversed,  the  wheel  will  return  as  before,  and  thus  the 
vibrations  be  continued  forever,  as  in  the  case  of  the  pendulum,  but  for 
the  waste  of  living  force  from  friction,  atmospheric  resistance,  and  absence 
of  perfect  elasticity  in  the  spring. 

9.  —  The  angular  acceleration  impressed  upon  the  balance  by  the  spring 


APPENI/IX   II.  241 

is  measured  by  the  moment  of  its  elastic  force  divided  by  the  moment  of 
inertia  of  the  entire  balance.  When  the  temperature  is  increased,  the 
spring  is  lengthened  and  the  elastic  force  it  exerts  lessened ;  the  wheel  is 
expanded,  its  matter  thrown  further  from  the  axis  of  motion,  and  the  mo- 
ment of  inertia  consequently  increased.  On  both  accounts  the  angular 
acceleration  is  diminished,  and  the  balance  will  vibrate  slower,  and  the 
intervals  'between  the  checks  be  increased.  The  effect  is  just  reversed  when 
tho  temperature  is  diminished.  This  is  the  source  of  greatest  difficulty 
with  all  portable  time-keepers,  and  renders  the  common  watch  worthless 
for  any  thing  beyond  an  approximate  indicator  of  the  time. 

10.  —  To  remedy  this  defect,  the  common  wheel  is  replaced  by  what  is 
called  an  expansion  balance,  which  is  re-  Figi9- 
presented  in   the   figure.     A  A  is  a  bar 

which  receives  the  end  of  the  arbor  into 

an  aperture  at  its  middle  point.     To  the 

ends  of  the  bar  are  securely  attached  two 

compound  metallic  curves  C  C,  composed 

of  two  concentric  strips,  one  of  steel  and 

the  other  of  brass,  the  latter  being  on  the 

convex  side ;  these  are  soldered  or  burned 

together   throughout   their  entire  length. 

Each  of  these  curved  pieces  carries  a  heavy  mass  D  I),  movable  from  one 

end  to  the  other,  but  capable  of  being  secured  in  any  one  place  by  means 

of  a  small  clamp-screw  shown  in  the  drawing. 

Now  when  the  temperature  increases,  the  exterior  brass  expanding  more 
than  the  interior  steel,  the  ends  C  C  are  thrown  inward  towards  the  arbor, 
while  the  ends  of  the  bar  are  thrown  outward,  but  through  a  much  less 
distance  ;  and  thus  by  properly  adjusting  the  places  of  the  masses  D  D,  the 
moment  of  inertia  of  the  balance  may  be  made  to  vary  directly  as  the 
moment  of  the  elastic  force  of  the  spring ;  in  which  case  the  angular  ac- 
celeration becomes  constant,  and  the  intervals  between  the  interposition  of 
the  checks  equal. 

11.  —  To  regulate  the  rate,  two  large-headed  screws  B  B,  calk*!  mean- 
time screws,  are  inserted,  one  into  each  end  of  the  bar.     If  the  chronometer 
run  too  slow,  the  moment  of  inertia  is  too  great  for  that  of  the  spring,  and 
these  screws  must  be  screwed  up,  which  has  the  effect  to  lessen  the  dis- 
tance of  their  heads  from  the  axis  of  motion,  and  thus  to  lessen  the  mo- 
ment of  inertia,  and  increase  the  angular  acceleration.     If  the  chronometer 
run  too  fast,  the  screws  must  be  unscrewed,  the  effect  of  which  must  be 
obvious. 

16 


SPHERICAL  ASTRONOMY. 


12. —  The  escapement  is  of  the  kind  usually  called  the  detc  ched,  from 
the  fact  that  except  at  certain  instants  of  time,  the  whole  appendage  of 
the  balance-spring  is  relieved  from  the  action  of  the  scape-wheel. 


The  scape-wheel  is  represented  at'Jkf ;  it  is  urged  by  the  motor,  acting 
through  the  wheel-work,  to  move  in  the  direction  of  the  arrow-head.  A 
steel  roller  (7,  called  the  main  pallet,  is  firmly  fixed  to  the  arbor  of  the  bal- 
ance. In  the  pallet  is  a  notch  i,  having  one  of  its  faces  considerably  un- 
dercut, and .  covered  with  an  agate  or  ruby  plate  to  receive  the  action  of 
the  teeth  of  the  scape-wheel.  Securely  fixed  to  one  of  the  frame-plates  of 
the  chronometer  is  a  stud  B,  and  to  this  is  attached  a  spring  A,  called  the 
detent ;  this  spring  is  extremely  thin  and  weak  at  the  stud  B.  Attached 
to  the  detent  is  a  stud  D.  A  ruby  pin  projects  from  the  detent  at  c,  which 
receives  a  tooth  of  the  scape-wheel  when  one  escapes  from  the  pallet  bear- 
ing i.  From  the  stud  D  proceeds  a  very  delicate  spring  E,  called  the  lifting 
spring,  which  rests  upon  and  extends  beyond  a  projection  F  from  the  end 
of  the  detent ;  this  projection  being  so  made  that  the  lifting  spring  cannot 
move  in  the  direction  from  the  scape- wheel  without  taking  the  detent  with 
it,  and  thus  lifting,  as  it  were,  the  pin  c  from  the  tooth  with  which  it  is 
in  contact,  while  it  leaves  the  lifting  spring  free  to  move  towards  the  scape- 
wheel  without  disturbing  the  detent.  Concentric  with  the  main  pallet,  at- 
tached to  and  just  above  it,  is  a  small  projecting  stud  a,  called  the  lifting 
pallet,  which  is  flattened  on  the  face  turned  from  the  scape-wheel  and 
rounded  on  the  other.  The  flattened  is  called  the  lifting  face. 

13.  —  Mode  of  Action. — In  the  position  of  the  figure,  the  main  pallet, 
under  the  action  of  the  balance-spring,  is  moving  in  the  direction  of  the 
arrow-head/,  and  the  lifting  pallet  is  coming  .with  its  lifting  face  in  contact 
with  the  lifting  spring  E,  which  it  lifts  with  the  detent  so  as  to  raise  the 
pin  c  clear  of  the  tooth  of  the  scape-wheel  with  which  it  i?  in  contact, 


APPENDIX    II.  243 

By  the  time  the  wheel  is  free  from  the  pin  c,  the  main  pallet  has  advanced 
far  enough  to  receive  an  impulse  from  the  "tooth  .t  upon  its  jewelled  surface. 
ij  and  before  this  tooth  escapes,  the  lifting  pallet  a  pails  with  the  lifting 
spring  E,  and  the  detent  returns  to  its  place  of  rest  and  interposes  the  pin 
c  to  receive  the  tooth  t(  as  soon  as  the  tooth  t  has  been  liberated  by  the 
onward  movement  of  the  main  pallet  from  its  face  i.  The  balance  having 
performed  a  vibration  ty  the  impulse  given  to  the  main  pallet,  returns  by 
the  action  of  the  balance-spring,  and  with  it  the  lifting  pallet  0,  whose 
rounded  face,  pressing  against  the  lifting  spring  E,  raises  it  and  passes,  first 
the  detent  without  disturbing  the  latter,  then  the  lifting  spring,  and  moves 
on  till  the  balance  has  completed  the  vibration,  when  it  returns  to  the  po- 
sition indicated  in  the  figure,  and  the  same  evolution  is  performed  again  ; 
the  balance  thus  making  two  vibrations  for  every  impulse. 

The  Vernier. 

1.  —  This  is  a  device  by  which  the  value  of  any  portion  of  the  lineal 
distance  between  two  divisions  of  a  graduated  scale  of  equal  parts  may  be 
found  in  terms  of  the  space  itself. 

It  consists  of  a  scale  whose  length  is  equal  to  any  assumed  number  oj 
parts  of  that  to  be  subdivided,  and  is  divided  into  equal  parts  of  which  the 
number  is  one  greater  or  one  less  than  the  number  of  the  primary  scale 
taken  for  the  length  of  the  vernier. 

A  ^-T-I  t  ?  I--HPB 


11     I     If 


Let  AD  be  any  scale  of  equal  parts,  and  denote  by  s  the  length  of  n—  1 
of  these  parts ;  then  will 


n  —  1 


be  the  value  of  the  unit  of  the  scale.     Take  a  vernier  B  E  of  equal  length 
s,  and  suppose  it  divided  into  n  equal  parts,  then  will 


be  the  length  of  one  of  its  parts,  and  the  difference  of  length  between  m 
parts  of  the  scale  and  an  equal  number  of  parts  of  the  vernier,  will  be 

ms        ms          m  .s 
n^l  ~  V  =  n  »-!' 


244  SPHERICAL    ASTRONOMY. 

But is  the  value  of  the  unit  of  the  scale,  and  n  the  whole  number  of 

n — 1 

divisions  of  the  vernier ;  denoting  the  first  by  F,  this  difference  may  be 
written 

-  F. 

n 

ISTow,  the  length  of  a  part  on  the  scale  is  greater  than  that  on  the  vernier, 
and  the  number  of  parts  on  the  vernier  is  greater  by  one  than  the  number 
in  an  equal  length  on  the  scale ;  hencer  if  the  mth  intermediate  division  of 
the  vernier  coincide  with  any  one  division  on  the  scale,  the  zero  of  the  ver- 
nier will  fall  between  two  divisions  of  the  scale,  and  be  in  advance  of  that 
bearing  the  smaller  figure  by  the  distance  expressed  above ;  so  that,  taking 
the  zero  of  the  vernier  as  the  index  or  pointer,  its  distance  from  the  zero  of 
the  scale  will  be  the  number  of  units  denoted  by  the  figure  on  the  division 

next  preceding,  plus  the  — th  part  of  the  unit  of  the  scale.     Thus,  in  the 

figure,  A  being  the  zero  of  the  scale,  B  that  of  the  vernier  and  therefore 
the  pointer,  the  distance  of  the  latter  from  the  former  will  be  Aa-{-aB', 
and  because  n—10,  and  the  division  b  of  the  scale  coincides  with  the  4th 

4 
of  the  vernier,  m=4,  and  the  distance  AB=Aa-] .fe. 

2.  —  The  least  value  that  may  be  read  with  certainty  is  obtained  by 
making  m=\,  which  will  give, 

F 

n'         » 

Whence  we  have  this  rule  for  finding  the  lowest  reading  by  means  of  the 
vernier,  viz. : 

Divide  the  lowest  count,  or  unit  of  the  scale,  by  the  number  of  divisions 
on  the  vernier. 

If  the  scale  be  tenths  of  inches,  and  we  make  w=10,  then  will 

7=^-10=i; 

in  which  case  the  subdivisions  will  be  carried  to  hundredths  of  inches. 

3.  —  The  vernier  is  equally  applicable  to  all  kinds  of  scales,  to  circular 
as  well  as  rectilinear :    the  only  condition  being  that  the  different  parts 
shall  be  equal. 

Suppose  each  degree  on  the  circumference  of  a  circle  is  divided  into  6 
ojual  parts,  and  that  tbo.  number  of  parts  on  the  vernier  is  60,  then  will 


APPENDIX   II.  245 

V=10' 
and 


n         60 

So  that  the  read'  ng  of  angles  with  an  instrument  having  such  a  circle  may 
be  carried  to  ten  seconds. 

Micrometer. 

1.  —  The  Micrometer  is    an  instrument   employed    to  make  minute 
measurements,  and  is  applicable  alike  to  time  and  linear  distance.     It  has 
various  forms. 

1*.  —  The  Reticle.  —  He  who  views  a  distant  object  through  a  telescope, 
does  not  look  at  the  object  but  at  its  image  within  the  tube  of  the  instru- 
ment. The  image  of  a  point  is  always  in  a  plane  through  the  focus  of  the 
lens  conjugate  to  the  point  itself,  and  perpendicular  to  the  tube  of  the  tel- 
escope. The  visible  portion  of  this  plane  is  called  the  field  of  view.  Some 
point  in  the  field  of  view  is  arbitrarily  assumed  as  an  origin  of  reference,  and 
marked  by  the  intersection  of  a  pair  of  cross  wires.  The  line  through  this 
point  and  the  optical  centre  of  the  field  lens,  is  called  the  line  of  collimation. 

2.  —  If  the  telescope  be  at  rest  and  an  object  in  motion,  the  image  of 
any  one  of  its  points  will  when  visible  pass  across  the  field  of  view  ;  and 
one  of  the  opaque  wires  being  made  to  coincide  with  its  path,  the  image 
will  move  directly  towards  the  line  of  eollimation,  and  ftie  exact  instant  of 
its  reaching  it  may  be  noted.     But  every  such  observation  is  liable  to  error. 
To  increase  the  chances  of  avoiding  this  error,  the  wires  marking  the  line 
of  collimation  are  made  perpendicular  to  one  another,  and  an  equal  number 
of  equidistant  and  parallel  wires  added  on  either  side  of  that  which  is  per- 
pendicular to  the  path  of  the  image.     When  the  motion  of  the  image  is 
uniform,  an  average  of  the  times  of  passing  the  parallel  wires  will,  accord- 
ing to  the  doctrine  of  chances,  give  a  time  of  passing  the  line  of  collima- 
tion more  free  from  error  than  the  single  observation. 

3.  —  This  simple  form 
of  the  micrometer  is  call- 
ed a  reticle.    The  wires  or 
spider  lines  are  stretched 
across  a  circular  metallic 
diaphragm  pierced  by  a 

large  concentric  opening.  On  the  edge  of  the  diaphragm,  and  in  the 
prolongation  of  the  single  wire,  two  studs  project  at  right  angles  to  its 
plane  ;  and  these,  with  two  antagonistic  screws  A  B,  hold  the  reticle  in  po- 


246  SPHERICAL   ASTRONOMY. 

sition  ;   the  screws,  for  this  purpose,  passing  through  the  tube  of  the  teles 
cope  and  leaving  the  heads  exposed  for  purposes  of  adjustment. 

4.  —  Position  Filar  Micrometer. — The  purpose  of  this  instrument  is 
to  measure  the  angles  at  the  observer,  subtended  by  the  distances  between 
objects  that  appear  very  close  together,  and  to  determine  the  positions  of 
the  planes  of  these  angles.     It  consists  of  two  parts,  viz. :   One  to  measure 
the  angle  between  the  objects ;  and  the  other,  the  inclination  of  the  plane 
:>f  the  objects  and  observer  to  some  co-ordinate  plane. 

5.  —  The  first  is  represented  in  the  figure,     a  and  c  are  two  fine  par- 
allel wires,  which  are  made  to  move  at  right  angles  to  their  lengths  by 
means  of  screws  firmly  connected  with  the  forks  A  and  (?,  to  whose  prongs 
they  are  attached.     The  screws  have  fifty  threads  to  the  inch,  and  are 

Fig.  12. 


moved  by  nuts  sc  mounted  as  to  admit  of  a  motion  of  rotation  without 
translation,  so  that  by  turning  the  nuts  a  motion  of  translation  is  commu- 
nicated to  the  wires  in  either  direction,  depending  upon  the  direction  of 
the  rotation.  Thl  outer  surfaces  of  the  nuts  are  cylindrical,  and  enter  fric- 
tion tight  the  central  perforations  of  two  circular  wheels  whose  planes  aro 
perpendicular  to  the  lengths  of  the  screws,  and  which  are  large  enough  to 
admit  of  their  circumferences  being  divided  into  100  equal  parts,  which 
parts  are  marked  and  numbered.  Each  wheel  is  provided  with  a  station- 
ary pointer  or  index. 

A  third  and  stationary  wire,  perpendicular  to  the  first  two,  is  supported 
by  a  diaphragm  disconnected  from  the  forks.  Upon  one  of  the  interior 
edges  of  this  diaphragm,  and  parallel  to  its  wire,  is  a  graduated  scale  in 
the  shape  of  a  comb,  having  50  teeth  to  the  inch,  so  that  one  revolution  of 
a  nut  will  carry  its  movable  wire  from  the  centre  of  one  valley  between  the 
teeth  to  that  of  the  next.  Near  the  central  valley  of  the  scale  is  a  small 
hole  to  mark  the  zero  of  the  comb-scale,  from  which  the  scale  is  estimated 
in  either  direction.  It  is  easily  seen  that  a  turn  of  the  nut-head  through  one 
of  its  divisions  will  move  its  wire  through  a  linear  distance  eqiuJ  to  y-j^  of 
s*o  or  5^  oW  °f  an  m°h »  and  having  ascertained  by  the  measurement  of 
some  small  distance  on  the  circumference  of  a  great  circle  of  the  celestial 
sphere,  or  by  the  process  in  Example,  p.  249,  its  equivalent  in  arc,  thi?.  the 


APPENDIX   II.  247 

micrometer  part  of  the  arrangement,  is  readily  applied  to  the  dete  'mi  nation 
of  small  angles. 

6.  —  The  second  and  position  part  consists  of  a  circular  plate  A  A, 
called  the  position  circle,  some  three  or  four  inches  in  diameter,  having 
its  circumference  divided  into  360°,  which  are  again  subdivided  to  any 
convenient  extent.  The  central  part  is  cut  away,  and  the  micrometer 
arrangement  so  attached,  with  its  wL  es  parallel  to  the  position  circle,  as 
to  admit  of  a  free  motion  Fig.  ia 

of  rotation  about  an  axis 
through  its  centre,  and  per- 
pendicular to  the  plane  of 
the  wires.  To  the  revolving 
plate  of  the  micrometer  part 
are  attached  two  verniers 
V  V,  and  motion  is  com- 
municated to  the  latter  by  a 
ratchet  and  pinion,  of  which 

latter  the  head  is  seen  at  0.  The  microscope  by  which  the  wires  and 
comb-scale  are  magnified,  and  which  serves  also  for  the  eye-glass  of 
the  telescope,  is  represented  at  E.  By  means  of  a  screw  cut  upon  a 
projecting  ring  .around  the  large  and  central  aperture  of  the  position  circle, 
the  instrument,  as  represented  in  the  figure,  is  attached  to  the  tail  end  of 
the  telescope. 

7. — To  measure  the  angular  distance  between  two  objects  in  the 
field  of  view,  turn  the  head  0  till  the  fixed  wire  passes  through  their 
images,  then  bisect  the  images  by  the  movable  wires ;  note  the  reading 
on  the  comb-scale  and  upon  the  heads;  take  their  sum  or  difference 
according  as  the  wires  are  on  opposite  sides,  or  same  side  of  the  zero  of 
the  comb-scale.  This  reduced  to  arc  will  be  the  measure  sought.  Note 
also  the  reading  of  the  position  circle ;  this  will  give  the  inclination  of  the 
plane  of  the  angle  to  the  plane  through  the  zero  of  the  position  circle.  A 
second  angle  being  measured  in  the  same  way,  the  difference  between  the 
second  and  first  reading  of  the  position  circle  will  give  the  inclination  of 
the  planes  of  the  two  angles. 

Micrometer  Revolution. 

THE  micrometer  being  supposed  ir  place,  and  the  eye-piece  pressed  for- 
ward far  enough  to  obtain  a  distinct  view  of  the  wires,  the  telescope  is 
directed  to  some  distant  object,  and  adjusted  to  distinct  vision.  An  image 
of  the  object  will  be  formed  on  the  plane  of  the  wires,  and  any  one  of  its 


24:8  SPHERICAL    ASTRONOMY. 

linear  dimensions  may  be  measured  by  turning  the  position  circle  till  the 
stationary  wire  coincides  with,  and  the  movable  wires  pass  through  the 
extremities  of  its  image.  The  number  of  entire  comb-teeth  between  the 
movable  wires,  multiplied  by  100,  and  this  product  increased  by  the  sum 
of  the  readings  of  the  screw-heads,  will  give  the  linear  dimensions  of  the 
mage  expressed  in  units  of  the  screw-head.  The  value  of  the  latter  is,  in 
/he  case  we  have  taken,  g-^-Q"  °^  an  mcn-  To  find  the  angle  subtended 
by  the  object,  we  must  know  the  angular  value  of  the  unit  on  the  screw- 
head. 

It  is  demonstrated  (Optics,  §  60)  that  the  optical  image  of  any  point  of 
an  object,  is  on  a  right  line  drawn  through  the  point  and  the  optical  cen- 
tre of  the  lens  by  which  the  image  is  formed.  The  angles,  at  the  optical 
centre,  subtended  by  an  object  and  its  image,  are  therefore  equal,  and  if 
the  images  of  objects  which  subtend  equal  angles  were  at  the  same  dis- 
tance from  the  optical  centre,  they  would  be  of  the  same  size.  The  lineai 
dimensions  of  the  images  at  the  same  distance  from  the  optical  centre, 
would  therefore  be  proportional  to  the  angles  subtended  by  their  respective 
objects,  and  to  find  the  angular  value  in  question,  it  would  be  sufficient 
to  cause  the  image  of  some  well-defined  object,  whose  distance  and  dimen- 
sions are  known,  to  be  embraced  by  the  wires,  and  to  divide  the  angle  which 
the  object  subtends,  expressed  in  seconds,  as  determined  trig&nometrically, 
by  the  number  of  units  of  the  screw-heads,  which  indicate  their  separation. 
But  the  distances  and  therefore  the  dimensions  of  images,  whose  objects 
subtend  the  same  angle,  are  variable,  being  dependent  on  the  distance  of 
the  objects,  and  from  the  value  found  by  the  above  process  must  be  de- 
duced that  which  would  have  resulted  had  the  image  been  formed  at  some 
constant  distance,  which  is  that  of  the  principal  focus. 

Let  /  and  /"  denote  the  distances  respectively  of  the  object  and  its 
image  from  the  optical  centre,  and  Fu  the  principal  focal  distance  of  the 
object-  glass,  supposed  convex.  Then,  Optics,  §  44,  Eq.  (40), 

fa-          ~ 
' 


/"'         J 

and  denoting  by  n  and  JV,  the  number  of  units  of  the  screw-heads  when 
the  image  is  embraced  at  the  distances  f  and  Fn  respectively,  we  shall 
have,  Optics,  §  64,  Eq.  (58), 

/"  :  Fu  :  :  n  :  JIT; 
whenca 

F  f—F 

*•=*«..,,-_=_„/__  M 


APPENDIX    II.  249 

and  calling  a,  the  number  of  seconds  in  the  angle  subtended  by  the  object^ 
we  have,  by  the  rule  just  given. 


JSxample.  —  The  length  of  the  object  measured  in  a  direction  perpen- 
dicular to  the  line  of  sight  was  3  feet  ;  the  distance  from  the  object-glass, 
261.9  yards  ;  the  principal  focal  length,  45.75  inches;  and  the  sum  of  the 
divisions  on  the  screw-heads  indicating  the  separation  of  the  wires,  1819. 
Then 

/=  261.9yd8-;  Fn  =  45.75in-  =  1.2708yds-;  n=  1819. 
f—Fn  —  260.6292yd3-. 

R    0  5yds- 
tan  -i  a  =      '    '  -g-,  of  which  the  log.  is  7.280835  ; 

whence 

a=  13'  07".57  =  787".57. 

Log.  a         ....  2.8962892 

"     /          .         .         .         .  2.4181355 

•'      n         a  comp.      .         .  —4.7401673 

"         —  Fu  "               .         .  —3.5839923 


-1.6385843 


Now,  to  measure  the  angle  subtended  by  the  distance  between  any  twe 
points,  direct  the  telescope  so  as  to  get  the  images  of  the  points  in  the 
field,  and  turn  the  micrometer  till  the  stationary  wire  apparently  passes 
through  them,  and  by  a  motion  of  the  screw-heads  bring  the  movable 
wires  to  the  images  —  the  number  of  units  of  '-he  screw-head,  which  indi- 
cate the  separation  of  the  wires,  multiplied  by  the  decimal  0".4351,  will 
give  the  number  of  seconds  in  the  angle. 

The  value  of  —  ,  being  a  function  of  Fit^  Eq.  (a),  will  of  course  vary 

with  the  object-glass,  but  is  perfectly  independent  of  the  eye-glass. 

If  the  distance/  be  so  great  that  Fn  may  be  neglected  in  comparison, 
then  will  Eq.  (a)  give 

N  —  n, 

which  will  be  the  case  when  the  angular  value  is  determined  from  astro- 
nomical objects. 


250  SPHERICAL   ASTRONOMY. 

Spirit-Level. 

1.  —  This  is  an  instrument  used  to  adjust  a  line  to  a  given  position  in 
reference  to  the  horizon. 

It  consists  of  a  cylindrical  glass  tube  A  A,  whose  axis  is  the  arc  of  a 
circle.  This  tube  is  filled  nearly  full  with  some  one  of  the  more  perfect 
fluids,  such  as  alcohol  or  naphthalic  ether,  leaving  a  small  portion  of  air, 
seen  at  B,  called  the  air-bubble,  and  hermetically  sealed  at  both  ends.  It 


Fig.  14. 


is  then  usually  set  in  a  metallic  tube  (7,  very  much  cut  away  on  one  side 
from  the  middle  towards  the  ends,  so  as  to  exhibit  the  bubble  and  fluid 
when  in  a  horizontal  position.  This  metallic  tube  is  connected  with  a 
plate  of  metal  F  F,  by  a  hinge  E  and  screw  Z>,  the  axis  of  the  hinge 
being  perpendicular,  and  that  of  the  screw  parallel  to  the  plane  of  the 
circular  axis  of  the  level. 

2.  —  A  scale  of  equal  parts  is  cut  either  upon  the  upper  surface  of  the 
glass  tube  or  upon  a  slip  of  ivory  and  metal  lying  in  the  plane  of  the  tube's 
curve,  as  represented  at  G  G.     The  divisions  of  the  scale  being  numbered, 
the  value  of  the  spaces  in  arc  is  readily  ascertained  by  attaching  the  level  to 
the  face  of  a  vertical  graduated  circle,  and  turning  the  latter  sufficiently  to 
cause  the  air-bubble  to  pass  from  one  end  of  the  scale  to  the  other.'     The 
angular  space  passed  over  by  the  circle  reduced  to  seconds,  divided  by 
the  number  of  units  on  the  scale  traversed  by  the  bubble,  will  give  the 
value  of  the  unit  in  some  multiple  of  the  second. 

3.  —  Use. — The  surface  of  the  fluid  being  always  horizontal,  the  line 
connecting  the  ends  of  the  bubble  will  be  a  level  chord  of  the  level's  arc, 
and  the  radius  passing  through  the  point  of  the  scale  midway  between  the 
ends  of  the  bubble  will  be  vertical. 

Now,  suppose  any  line  of  an  instrument  with  which  the  level  is  used 
to  be  made  parallel  either  to  the  radius  passing  through  the  zero  of  the 
scale,  or  to  the  chord  whose  ends  are  marked  by  the  same  numbers ; 
then,  to  make  this  line  vertical  in  the  first  case,  or  horizontal  in  the 
second,  move  the  instrument,  the  level  being  securely  attached,  till  the 
ends  of  the  bubble  are  equally  distant  from  the  zero. 

If  the  ends  of  the  bubble  be  not  at  the  same  distance  from  the  zero, 
the  inclinat'on  x  of  the  line  in  question  to  the  vertical  cr  horizontal 


APPENDIX   II.  251 

direction  is  thus  found  :  Let  a   denote  the  semi-length  of  the  bubble, 
m  and  n  the  numbers  of  the  scale  at  its  extremities,  then  will 

whence 

m  —  n 


This  value  of  x  being  independent  of  the  length  of  the  bubble,  which 
is  indeed  a  variable  quantity,  even  in  the  same  level,  because  of  its  varying 
temperature,  gives  the  inclination  of  the  line  under  consideration  to  its 
proper  position,  when  the  level  is  adjusted  to  the  instrument. 

If  the  lower  surface  of  the  plate  F  F  be  parallel  to  the  chords  of  equal 
numbers,  the  inclination  of  any  given  line  or  plane  may  be  ascertained  by 
laying  this  plate  upon  it  and  applying  the  above  rule. 

But  if  the  lower  surface  of  the  plate  be  not  parallel  to  the  chords  of 
equal  numbers,  its  inclination  to  them,  and  that  of  the  plane  or  line  in 
question  to  the  horizontal  or  vertical  direction  may  nevertheless  be  found 
thus  :  Denoting  the  first  by  y,  and  the  latter,  as  before,  by  .r,  and  using 
the  notation  of  equation  (2),  we  have  for  one  position  of  the  level, 


and  for  the  reversed  position  of  the  plate  with  its  level, 

whence 

_  l+l'  _  m—n+m'—n' 


I— I'      m  —  n—m'—n' 


If  the  given  surface  or  line  be  provided  with  adjusting  screws,  as  is 
the  case  in  all  astronomical  instruments,  the  ends  of  the  bubble  may  be 
brought  to  the  same  reading  in  the  first  position  of  the  level,  in  which 
case,  we  have  m=n,  and 

m'  —  n' 

*  =  —~=-y     .......    (3) 


The  angle  y  is  called  the  error  of  the  level,  and  the  angle  x  the  error 
in  level  of  the  instrument,  and  the   above  equation  gives  this  rule  for 
and  correcting  these  errors,  viz.  : 


252  SPHERICAL   ASTRONOMY. 

The  level  beinv  placed  over  the  given  line,  bring,  by  means  of  the 
adjusting  screws  of  the  instrument,  the  bubble  to  read  the  same  at  both 
ends  ;  then  reverse  the  level,  or  turn  it  end  for  end,  and  take  one  fourth 
of  the  difference  of  the  new  readings  ;  add  this  to  the  lesser  of  the  read- 
ings, and  turn  the  screw  D  till  the  end  of  the  bubble  nearest  the  zero 
reach  the  numb*  r  answering  to  this  sum,  to  which  add  again  the  same 
quantity,  and  bring  the  end  of  the  bubble  to  this  new  reading  by  the 
adjusting  screws  of  the  instrument.  The  ends  of  the  bubble  will  stand  at 
the  same  numbers,  and  both  errors  will  be  destroyed. 

Reading  Microscope. 

1.  —  This  instrument,  like  the  vernier,  has  for  its  object  to  read  and 
subdivide  the  space  between  two  consecutive  divisions  of  any  scale  of 
equal  parts,  and  is  the  most  perfect  yet  devised  for  this  purpose. 

It  is  a  compound  microscope,  whose  object-glass  forms  an  enlarged 
image  of  the  space  to  be  divided.  This  image  is  thrown  upon  the  plane 
of  two  spider-lines  or  wires,  arranged  in  the  form  of  a  St.  Andrew's  cross, 
and  so  placed  that  a  line  bisecting  its  smaller  angles  is  parallel  to  the 
cuts  or  division  marks  of  the  scale.  The  cross  is  attached  to  a  diaphragm, 
which  is  moved  by  a  micrometer  screw  in  the  direction  of  its  plane,  per- 
pendicular to  the  axis  of  the  microscope.  The  head  of  the  screw  is 
divided  into  any  number  of  equal  parts,  depending  upon  the  nature  of 
the  scale  and  the  extent  to  which  the  subdivisions  are  to  be  carried.  The 
numbers  on  the  head  are  so  placed  that  when  the  screw  is  turned  in  the 
direction  to  bring  them  in  the  order  of  their  increase  to  a  fixed  pointer, 
the  cross  shall  move  along  the  image-scale  in  the  direction  in  which  its 
numbers  decrease. 

Within  the  barrel  of  the  microscope  is  a  stationary  comb-scale,  like  that 
in  the  position  micrometer.  Its  plane  is  parallel  to  that  of  the  cross,  and 
the  distance  between  the  centres  of  two  valleys,  separated  by  a  single  tooth, 
is  equal  to  the  space  over  which  the  cross  is  moved  by  a  single  revolution 
of  the  screw.  Every  fifth  valley  is  cut  deeper  than  the  others  to  facilitate 
the  reading ;  and  near  the  bottom  of  the  central  valley  of  the  comb  is 
a  small  circular  aperture,  to  mark  the  zero  position  of  the  pointer  or 
index,  which  is  a  small  wire  attached  to  the  movable  diaphragm,  and  so 
placed  that  its  prolongation  shall  bisect  the  smaller  angles  of  the  cross. 

In  (1),  A  A  is  the  main  tube  of  the  microscope,  passing  through  a  collar 
or  support  B,  where  it  is  firmly  held  by  two  milled  nuts  g  g,  which  act 
upon  a  screw  cut  upon  the  outer  surface  of  the  tube.  These  nuts  also 
serve  to  change  the  distance  of  the  whole  microscope  from  the  scale  to 


APPENDIX  II. 


253 


be  read  ;  h  is  the  object-glass  placed  in  a  smaller  tube,  upon  whose  outer 
surface  is  also  a  screw,  by  which  this  glass  may  be  moved  independently 


Fig.  15. 


Fig.  16.     2. 


of  the  main  tube ;  the  diaphragm  of  the  cross  is  in  a  working  box,  whose 
edge  is  seen  at  a  ;  e  is  the  graduated  head,  firmly  attached  by  a  friction 
clamp  to  the  nut  b  of  the  micrometer  screw ;  /  is  a  pointer  attached  to 
the  working  box  ;  d  is  the  eye-glass,  which  moves  freely  in  the  direction 
of  the  axis  of  the  microscope  by  a  sliding  tube  ;  at  c'  is  represented  the 
head  of  a  small  screw,  which  supports  and  gives  motion  to  the  comb-scale 
within  the  working  box,  and  S  S  represents  the  edge  of  the  scale  to  be 
subdivided.  In  (2)  is  represented  the  field  of  view,  as  seen  when  the  eye 
is  applied  at  e?,  in  which  m  m'  is  the  image  of  the  scale,  with  one  of  its 
cuts  bisecting  the  smaller  angles  of  the  cross,  and  e  the  wire  index  at 
its  zero  position,  as  indicated  by  its  being  seen  through  the  centre  of  the 
circular  aperture  of  the  comb.  In  this  position  of  the  pointer,  the  zero 
of  the  graduated  head  e  is  brought  to  the  index  /,  by  holding  the  nut  b 
firmly  in  the  hand,  and  turning  the  head,  which  is  only  held  in  its  place, 
as  before  stated,  by  the  action  of  the  friction  nut. 

2.  —  The  quotient  arising  from  dividing  the  length  of  the  image  space 
by  that  over  which  the  wires  move  in  one  revolution  of  the  screw-head, 
as  given  by  the  comb-scale  and  head,  is  called  the  run  of  the  micrometer. 
For  convenience,  the  run  should  be  an  entire  number. 

3. —  The  image-scale  must  be  accurately  in  the  plane  of  the  wires, 
otherwise  there  would  be  a  parallactic  motion,  which  would  shift  the 
position  of  the  wires  on  the  image-scale  at  every  change  in  the  position 
of  the  eye,  and  thus  vitiate  the  measurement.  This  parallactic  motion 
is  easily  detected  by  slightly  shifting  the  position  of  the  eye  when  looking 
through  the  eye-glass. 

There  are,  then,  two  adjustments  for  the  reading  microscope,  viz.,  that 
for  the  run  and  that  for  parallax 


254  SPHERICAL    ASTRONOMY. 

4.  —  The  size  of  the  image  of  an  object,  and  its  distance  from  the 
lens  by  which  it  is  formed,  are  dependent  upon  the  distance  of  the  object 
from  the  lens,  being  greater  in  proportion  as  this  distance  is  less,  and  less 
HS  it  is  greater. 

If  the  distance  of  the  object-glass  of  the  microscope  from  the  scale 
be  changed  by  means  of  the  screw  on  the  tube  at  A,  the  size  of  the  image 
space  will  be  altered,  and  may,  therefore,  be  made  of  such  dimensions 
that  the  cross  will  move  from  one  division  to  the  next  in  order,  by  a  given 
entire  number  of  revolutions ;  and  if  by  this  operation,  the  image  be 
thrown  off  the  plane  of  the  wires,  as  it  in  general  will,  it  is  restored  by 
changing  the  distance  of  the  whole  body  of  the  microscope  from  the 
scale  by  means  of  the  milled  nuts  g  g.  By  two  or  three  efforts  cautiously 
conducted,  the  adjustments  may  be  made  without  difficulty. 

To  illustrate,  let  the  scale  be  that  of  the  sexagesimal  division  of  the 
circle,  and  suppose  each  degree  divided  into  twelve  equal  parts,  each 
space  will  be  equal  to  five  minutes ;  if  we  make  the  run  five,  each  tooth 
on  the  comb  will  be  equal  to  one  minute,  and  if  the  screw-head  be  divided 
into  sixty  equal  parts,  each  of  its  spaces  will  be  equal  to  one  second;  so 
that  the  circle  may  be  read  to  seconds. 

Now  suppose  on  examining  the  run,  which  is  done  by  turning  the 
screw-head  till  the  cross  moves  from  one  division  to  the  next  in  order,  it 
be  found  5'  10"  ;  it  is  too  great.  Move  the  object-glass  h  from  the  plane 
of  the  circle  by  screwing  in  its  tube,  the  image  will  decrease,  and,  if  it 
were  before  on  the  plane  of  the  wires,  it  will  now  pass  to  some  position 
between  that  plane  and  the  object-glass  h.  Move  the  whole  body  of  the 
microscope  by  means  of  the  milled  nuts  gg  towards  the  circle  ;  the  image 
will  be  restored  to  its  proper  position,  with  less  dimensions  than  it  had 
before.  By  one  or  two  repetitions  of  this  process  the  adjustments  are 
made. 

5.  —  The  wire  pointer  at  its  zero  position  on  the  comb-scale  is  the 
index  of  the  circle  or  instrument  scale.     When  the  pointer,  in  this  po- 
sition, is  immediately  opposite  a  division  mark  of  the  circle  scale,  say  the 
third  after  that  marked  27°,  which  is  indicated  by  the  angles  of  the  cross 
being  bisected   by  the   image   of    that    division   mark,   the   reading   is 
27°  15'  00" ;  but  if  the  intersection  of  the  cross  wires  falls  between  the 
third  and  fourth  divisions  after  that  marked  27°,  then  will  the  reading  be 
greater  than  that  above  by  the  value  of  the  distance  from  the  cross  wires 
to  the  division  mark  to  which  tho  cross  will  move  by  turning  the  screw- 
head  in  the  order  of  its  increasing  numbers.     To  find  this  value,  turn  (4ie 
screw-head  in  the  direction  just  indicated  till  'he  angles  of  ibe  cms?  ar* 


APPENDIX   II. 


255 


bisected  by  the  division  mark  in  question,  and  count  the  entire  number 
of  comb  teeth  between  the  aperture  and  pointer,  then  note  the  reading 
on  the  screw-head;  suppose  the  former  to  be  3  and  the  latter  41,  the 
true  reading  will  be27°18'41". 

The  Transit. 

1.  —  The  transit  is  an  instrument  which  is  used  in  connection  with 
a  time-piece  to  ascertain  the  precise  instant  of  a  body's  passing  the  me- 


256  SPHERICAL    ASTRONOMY. 

ridian  of  a  place. '  It  consists  of  a  telescope  T  T,  usually  of  considerable 
power,  permanently  fixed  to  a  substantial  axis  A  A,  at  right  angles  to  its 
length.  The  axis  terminates  at  each  end  in  a  steel  pivot,  accurately 
turned  with  a  diamond  point,  to  a  cylindrical  shape.  The  pivots  are  of 
equal  diameters,  received  into  notches  cut  in  two  blocks  of  metal,  called 
Ys,  which  rest  in  metallic  boxes,  the  latter  being  imbedded  in  metallic  or 
stone  piers,  according  as  the  instrument  is  intended  to  be  portable  or  fixed. 

2.  —  Permanently  attached  to  the  tail  or  eye  end  of  the  telescope, 
on  opposite  sides,  are  two  small  graduated  circles,  called  finders.     The 
planes  of  these  circles  are  perpendicular  to  the  axis  of  the  transit,  and  each 
circle  has  an  index-arm,  which  carries  a  small  spirit-level  and  two  verniers, 
one  at  each  end.     The  index-arms  are  movable  about  the  centres  of  their 
respective  circles,  and  are,  as  well  as  the  axis  of  the  transit,  provided 
with  a  clamping  and  tangent  screw  arrangement,  thus  affording,  with  the 
aid  of  the  level  and  verniers,  the  means  of  giving  the  telescope  any  de- 
sired inclination  to  the  horizon. 

3.  —  At  the  solar  focus  of   the   object-glass  of  the   telescope   is  a 
reticle,Y\g.  11,  in  which  the  single  is  replaced  by  a  double  wire,  with  small 
interval,  and  so  placed  as  to  be  parallel  to  the  axis  of  the  transit.     These 
are  called  axis  wires.     Those   wires  of  the  reticle   which   are   at   right 
angles  to  these  are  called  the  normal  ivires.     To  the  fixed  wires  of  the 
reticle  a  movable  one  is  added  ;  it  is  always  parallel  to  the  normal  wires, 
indeed,  is  itself  a  normal  wire,  and  is  put  in  motion  in  the  direction  of 
the  axis  wires  by  means  of  a  micrometer  screw,  with  graduated  head, 
shown  at  m. 

4.  —  The  small  tube  containing  the  eye-piece  of  the  telescope  is 
attached  to  a  slid  ing-frame,  connected  with  a  screw  e,  by  which  the  eye- 
piece is  carried  from  one  side  of  the  field  of  view  to  the  other,  in  the 
direction  of  the  axial  wires. 

5.  —  The  axis  is  hollow  throughout,  and  the  pivots  are  perforated 
at  the  ends  to  admit  the  light  from  a  lamp  L,  supported  upon  one  of  the 
piers.     This  light  is  received  by  a  reflector  within  the  tube  of  the  tel- 
escope, and  inclined  to  its  axis  under  an  angle  of  45°,  and  is  reflected  to 
the  eye-glass,  thus  illuminating  the  field  of  view,  and  exhibiting  the  wires 
of  the  reticle.     The  reflector  is  perforated  by  an  elliptical  opening  in  its 
centre,  to  permit  the  direct  light  from  any  external  object  to  pass  freely 
to  the  eye  end  of  the  telescope.     When  the  illumination  is  through  the 
other  end  of  the  axis,  the  reflector  is  revolved  through  an  angle  of  90°, 
by  means  of  a  milled-headed  wire,  with  which  it  is  permanently  con- 
nected.    The  head  is  shown  at  r. 


APPENDIX   II. 


257 


Fig.  18. 


Fig.  19. 


Fig.  20. 


G.  —  The  boxes  which  support  the  Ys  are  large  enough  to  permii 
i  slight  play  in  the  latter;  one  in  a  horizontal,  Fig.  18,  and  the  other  in 
a  vertical  direction,  Fig.  19,  the  motions  being  effected  by  antagonistic 
screws.  By  the  first  of  these  motions, 
the  line  of  collimation  is  brought 
to  the  meridian,  after  the  rougher  ap- 
proximations to  that  plane  are  made 
by  other  means,  and  by  the  second 
the  axis  is  made  horizontal  by  the 
aid  of  a  large  and  delicate  spirit- 
level,  Fig.  20,  mounted  upon  in- 
verted Ys,  far  enough  apart  to  rest 
upon  the  pivots. 

Adjustments. 


\ 


7.  —  The  transit  is  adjusted  within  itself  when  its  line  of  collimation 
is  perpendicular  to  its  axis ;  and  it  is  in  position,  when  its  axis  is  perpen- 
dicular to  the  meridian.      Its  finders  are  adjusted,  if  the  air-bubbles  at 
their  levels  indicate  the   same  reading  at  both  ends,  when  the  verniera 
indicate  the  true  inclination  of  the  line  of  collimation  to  the  vertical  or 
horizon. 

8.  —  It  is  by  no  means  necessary,  or  even   desirable,  to  aim  at 
perfect  adjustment.     It  will,  in  general,  be  much    safer  to-  reduce  the 
errors  of  adjustment  to  narrow  limits,  then  to  determine  their  amount, 
and  eliminate  their  effect  from  observation,  in  the  manner  to  be  described 
presently. 

9.  —  Line  of  Collimation. — Direct   the  telescope   to  some   small, 
distant,  and  well-defined  terrestrial  object.     Bring  it  apparently  between 
the  horizontal  wires,  and  measure  its  distance  from  the  central  normal 
wire   by  means  of  the  micrometer  and  movable   wire;  denote  this  dis- 

17 


258  SPHERICAL    ASTRONOMY. 

tance  by  c'.  Lift  the  transit  from  its  Ys,  turn  the  axis  end  for  end,  and 
measure,  as  before,  the  apparent  distance  of  the  same  object  from  the 
middle  wire,  and  denote  this  distance  by  c".  Place  the  movable  wire  at 
the  distance,  of 

c'+c" 


on  the  side  of  the  object  from  the  middle  wire,  and  move  the  whole 
reticle  by  the  antagonistic  adjusting  screws,  which  lie  in  the  direction  of 
the  axial  wires,  till  the  object  appears  on  the  movable  wire  ;  the  line  of 
collimation  will  be  adjusted. 

10.  —  Error  of  this  adjustment.  —  If  n  denote  the  value  in  arc  of 
the  micrometer's  unit,  then  will  the  angle  which  the  line  of  collimation 
makes  with  its  proper  position,  before  moving  the  diaphragm,  be 


(4) 


and  the  line  of  collimation  will  describe,  when  the  telescope  is  moved, 
a  conical  surface,  whose  intersection  with  the  celestial  sphere  will  be  a 
small  circle. 

Example. — When  the  telescope  is  pointing  to  the  south,  let  the  middle 
wire  appear  to  be  326.3  revolutions  to  the  right  hand  of  the  object ; 
when  the  axis  is  reversed,  let  it  appear  318.7  to  the  right,  then  will 

326.3-318.7 

n  . =  c= 

and  if  one  revolution  of  the  micrometer  correspond  to  the  space  an  equa- 
torial star  would  pass  over  in  three  seconds  of  time,  then  will 

3s.Xl5 
«  =  -T_  =  0'.45) 

and 

c=3.8xO".45  =  l".7l. 

11.  —  The  axis. — This  must  first  be  levelled,  then  moved  in  azimuth 
till  it  is  perpendicular  to  the  meridian. 

Mount  the  level  with  its  inverted  Ys  upon  the  pivots,  bring  the  bubble 
to  the  same  reading  at  each  end  by  the  adjusting  screw  of  the  level ; 
reverse  the  level,  and  bring  the  bubble  again  to  the  same  readings — half 
by  the  screw  of  the  level  and  half  by  the  vertical  antagonist  screws  of 
the  Y,  which  admits  of  vertical  motion.  Repeat  the  operation  once  or 
twice,  and  the  thing  is  done 


APPENDIX    II.  259 

• 

12.  —  The  error  in  ills  adjustment. — After  the  first  approximation, 
denote  by  e',  e" ,  &c.,  the  reading  of  the  east  end  of  the  level;  by  w',  w", 
<fec.,  the  same  of  the  west  end,  and  let  the  parenthesis  denote  the  end  of 
the  axis  marked  by  some  peculiarity,  such  as  the  clamp,  or  illumination ; 
then  mounting  the  level  in  its  place,  and  writing  its  readings  in  any  one 
position  upon  the  same  horizontal  line,  we  may  have 

First  position  of  level  .     .     .     .     e' (wf) 

Level  reversed    .     .     .     .     .     .     e" (w") 


u  i*  f  e'+e" 

Half  sums  of      ..... 


2  2 

These  half  sums  are  the  readings  which  the  level  would  have  indicated  in 
both  positions  had  it  been  in  perfect  adjustment,  and 


the  error,  or  inclination  of  the  axis  to  the  horizon,  expressed  in  the  level's 
unit,  provided  its  pivots  be  of  the  same  size.  But  lest  there  may  be  a 
difference  in  the  pivots,  reverse  the  axis,  and  apply  the  level  as  before, 
and  we  may  have 

For  first  position  of  level     .     .     (e"r)        .    .     .    w'" 
Level  reversed       .....     (V'")       .     .     .    w"" 


Half  sums 


. 
2  2 

whence 


and 


f        ... 
IQ~ 

will  be  the  angle  which  the  axis  makes  with  the  line  whose  inclination  is 
given  in  equation  (5),  whence,  denoting  the  inclination  of  the  axis  to  the 
horizon,  or  the  angle  which  a  plane  perpendicular  to  the  axis  makes  with 
a  vertical  plane  at  right  angles  to  the  projection  of  the  axis,  on  the  hori- 
zon, by  I',  we  shall  have 


This  value  is  expressed  in  terms  of  the  level's  unit  ;  if  nf  denote  th« 


2(50  SPHERICAL    ASTRONOMY. 

value  of  this  unit  in  seconds,  we  shall  have,  representing  tl/3  angle  in 
seconds  of  arc  by  /, 

l=n'l'=n'(8±t) (8) 

The  value  of  t  for  the  same  axis  is  constant,  and  must  be  determined 
by  taking  a  mean  of  a  great  many  careful  observations.  If  it  be  positive, 
the  pivot  at  the  clamp  end  of  the  axis  is  the  larger,  but  if  negative,  it  is 
the  smaller. 

When  the  half  sum  of  the  readings  on  the  west  end  is  greater  than 
that  on  the  east,  the  inclination  is  counted  positive,  and  the  plane  perpen- 
dicular to  the  axis  .will  fall  to  the  east  of  the  zenith  ;  and  as  it  is  obvious 
that  the  axis  will  be  depressed  on  the  side  of  the  greater  pivot,  when  the 
level  indicates  perfect  adjustment,  the  upper  sign,  in  equation  (8),  must 
be  taken  when  that  pivot  is  to  the  east,  and  lower  when  to  the  west. 

Example. — Performing  the  operations  indicated,  let  the  following  be 
the  record,  viz. : 

First  position  of  level      .     .     .       71.40  (87.60) 

Level  reversed 78.60  (80.10) 

150.00  167.70 

167.70 


4)17.70(4.425=:  .s 
Axis  reversed. 

First  position  of  level     .     .     .       (73.95)  84.90 

Level  reversed     .....       (81.30)  77.85 

155.25         162.75 
162.75 

'4)7.50(1.875=s' 
Adding  the  indications  of  the  level  diagonally,  we  have 


16 

Applying  the  level  to  the  face  of  some  vertical  graduated  circle,  §  95, 
let  23.5  of  i*s  units  correspond  to  30"  then  will 

OA" 

n'=  —  =  1".276. 

23.5 

Whence  for 

Clamp  end  west  J=(4.425  —  0.fl37)Xl".276  =  4".83348 
Clamp  end  east  J=(1.875  +  0.637)xl    .276  =  3".205312 


APPENDIX   II. 

13.  —  Azimuth  adjustment. — It  is  now  supposed  that  tLe  errors  of 
collimation  and  of  level  are  destroyed.  By  a  reference  to  a  map  of  the 
Btars  it  will  be  seen  that  a  straight  line  drawn  from  the  Pole  star  to  a 
point  midway  between  the  fifth  and  sixth  stars,  called  s  and  £  respectively, 
in  the  constellation  of  the  Great  Bear,  will  pass  sensibly  over  the  pole. 
About  the  time  when  this  line  assumes  a  vertical  position,  direct  the  tel- 
escope to  the  Pole  star,  and  keep  its  image  on  the  middle  normal  wire 
by  a  motion  of  the  horizontal  adjusting  screws  of  the  Y,  or  by  the  mo- 
tion of  the  Ys  themselves,  if  the  requisite  range  be  beyond  that  of  the 
screws,  and  at  the  instant  when  it  is  inferred  from  a  suspended  plum- 
met, that  the  line  referred  to  is  exactly  vertical,  arrest  the  motion  and 
secure  the  Ys.  The  adjustment  will  be  sufficient  for  the  first  approx- 
imation. 

Next  find  the  amount  of  azimuth  error.  The  axis  being  horizontal,  and 
the  line  of  collimation  perpendicular  to  the  axis,  it  is  plain  that  in  the 
motion  of  the  telescope  the  line  of  collimation  will  describe  the  plane 
of  a  vertical  circle,  and  that  the  angle  made  by  this  plane  with  the  me 
ridian  is  the  error  in  question. 

Let  HOR  be  the  horizon,  RZH  Fig  21 

the  meridian,  P  the  pole,  Z  the  ze- 
nith, and  S  the  star  when  on  the 
line  of  collimation.  Make, 

X  =latitude  of  place=90°  —  ZP; 

S  =  declination  of  star  =  90°  —  PS, 
positive  when  north,  nega- 
tive when  south ; 

P=Z  P  £=hour  angle  of  star  ; 

Z  =ff  ZS= azimuth  of  star's  position,  and  equal  to  the  error  sought 
when  east, 

z  =ZS= zenith  distance  of  star. 

Then,  in  the  triangle  Z  P  S, 

.  sin  Z  .  sin  z 

sin  P— _ — 

cos  S 

and  because  the  sines  of  P  and  Z  are  very  small, 

^sinjX-,5)    z 

cos  6 

in  which  P  and  Z  are  expressed  in  seconds  of  arc. 


262  SPHERICAL   ASTRONOMY 

Diviie  both  members  by  15  and  make 


COS    0 

ind  equation  (9)  becomes 


p 

in  which  —  is  the  time  required  for  the  star  to  pass  from  the  vertical  de- 

scribed by  the  line  of  collimation  to  the  meridian,  and  if  t  denote  the  time 
indicated  by  a  timepiece  at  the  instant  the  star  is  on  the  central  normal 
wire,  the  time  of  meridian  passage  will  be 


Let  e  be  the  error  of  the  timepiece  at  the  time  t  referred  to  the  vernal 
equinox  ;  m  the  rate  or  quantity  by  which  this  error  is  increased  or  di- 
minished in  one  day  or  twenty-four  hours  ;  then,  if  R  denote  the  right 
ascension  of  the  star,  supposed  known,  will 


(12) 

1U 

and  for  a  second  star 

15 

in  which  t'  —  t  is  reduced  to  the  decimal  part  of  a  day.     Subtracting  the 
first  from  the  second,  we  get 

15 

in  which  the  upper  sign  is  used  when  the  timepiece  runs  too  slow,  and 
the  lower  when  too  fast ;  whence, 

....     (13) 

Z  is  hence  known,  and  for  which  the  instrument  may  be  corrected, 
if  desired.  This  value  in  equation  (11),  gives  the  time  of  meridian  pass- 
age, and  in  equation  (12),  which  may  be  written 


L  =  R-.T,      ......     (13') 


gives  the  error  of  the  timepiece. 


APPENDIX   II.  263 

The  sign  of  the  quantity  k  changes  when  ^he  declination  of  the  star 
exceeds  the  latitude,  and  also  when  the  star  passes  below  the  pole,  since 
in  this  latter  case  S  becomes  90°,  plus  the  polar  distance.  The  right 
ascension  for  all  stars  which  pass  below  the  pole  must  be  diminished  by 
twelve  hours. 

Using  the  Polar  star  in  its  upper  and  lower  passage  instead  of  two 
separate  stars,  equation  (13)  becomes 


15  ~  k'+k 


When  three  consecutive  transits  of  the  Pole  star  are  observed,  and  the 
intervals  are  equal,*  Z  will  be  zero,  and  the  transit's  axis  is  perpendicular 
to  the  meridian. 

The  values  of  k  and  k',  in  equation  (13),  must  be  found  from  stars 
differing  at  least  50°  in  declination. 

14.  —  Let  it  now  be  supposed  that  after  adjusting  the  transit  in  the 
manner  explained,  there  is  still  (as  in  general  there  will  be)  remaining  a 
su  all  error  in  collimation,  level,  and  azimuth.  It  remains  to  be  shown 
how  the  effects  of  these  may  be  eliminated  from  the  observation,  and  a 
result  obtained  the  same  as  though  the  instrument  had  been  perfect. 

Let  all  the  circles  referred  to  in  what 
precedes  be  projected  on  the  horizon, 
represented  in  M  ER  Q.  Let  Z  be  the 
zenith  ;  P  the  pole  ;  MZR  the  merid- 
ian ;  VZ  V  a  vertical  circle  at  right 
angles  to  the  projection  of  the  axis  of 
the  transit  ;  Vs'  V  the  circle  at  right 
angles  to  the  axis  ;  Cs  C'  the  parallel 
small  circle  cut  from  the  celestial  sphere 
by  the  motion  of  the  line  of  collima- 
tion ;  E  Q  the  equator;  and  esg  the 
diurnal  path  of  a  star. 

When  the  star  appears  on  the  central  normal  wire,  it  will  be  at  s  ;  and 
if  the  time  be  noted  and  increased  by  the  angle  s  P  0,  expressed  in 

J  O  .T 

time,  we  shall  have  the  indication  of  the  timekeeper  when  the  star  is  on 
the  meridian.     Now, 

s  P  0=sPs'+s'Ps"+s"P  0  ; 
the  angle  s  P  s'  is  measured  by  an  arc  of  the  equator,  which  is  equal  to 


264  SPHERICAL   ASTRONOMY. 

**'  divided  jy  the  cosine  of  the  distance  of  s  s'  from  the  equator,  which 
distance  is  the  declination  of  the  star.     But 


c  being  as  before  the  error  of  collimation  ;  hence, 

c 


sPs'  = 


COS 


The  angle  s'  Vs"  is  the  error  of  the  level  denoted  by  I.  Then  re- 
garding Ps"V  as  the  arc  of  a  great  circle,  from  which  it  will  differ  by 
an  inappreciable  quantity  within  the  limits  of  the  supposed  errors,  we 
shall  have,  in  the  triangle  Ps'V,  writing  the  small  angles  for  their  sines, 

'_7 

7  -  t>    . 

' 


*  -       •  ~~,  -    >    .    •  --  r  -  , 

sin  Ps'  cos  5 

representing  the  zenith  distance  by  (X—  £),  to  which  it  is  nearly  equal,  and 
regarding  Vs'  as  the  altitude,  from  which  it  differs  but  by  a  very  small 
quantity. 

The  angle  s"P  0  is  given  by  equation  (9),  Z  denoting  as  before  the 
azimuth  error.  Whence,  denoting  by  ~t  the  time  of  observation,  we  obtain 
for  the  time  of  meridian  passage 

__o_         J_    cosjX-5)       Z    sinjX-5) 
"^15.  cos  ^"^  15  '      cos  S  15*      cos  6 

in  which  c,  £,  and  Z  may  be  found  in  the  manner  already  indicated,  or 
still  better  as  follows. 
Making 


(16) 


and  supposing  the  timepiece  regulated  by  the  vernal  equinox,  and  rep- 
resenting its  error  at  the  time  t  by  e,  and  denoting  by  R,  the  right  ascen- 
sion of  the  star,  we  obtain 

t+e+c.C+l.L+Z.Z,=R (17) 

m  which,  if  e,  c,  /,  and  Z  be  regarded  as  unknown,  their  values  may  be 
found  by  carefully'  observing  four  stars,  whose  positions  are  well  known, 
and  which  differ  but  little  in  right  ascension,  and  considerably  in  declina- 


c  1 

1  5  .  cos  o 
cos(X—  S) 

15  .  cos  d 
sin(X-S) 

15  .cos  8        '  J 

APPENDIX   If.  265 

tion  The  values  of  (7,  Z,  and  Zt  being  computed  ill  each  case  from 
equation  (16),  we  may  have 

f     +e'+c.C'     +I.L'    +z.Z,'     =Rf 
t"    +ef+c.C"    +  I.L"  +Z.Z/'   =  R" 
t'"  +e'+c.  C'"  +  I.L'"  +z.ZS"  =R'" 
t""+ef+c  .  C""+l .  L""+z  .  Z,""=R"" 

which  are  sufficient.  But  as  there  are  always  slight  errors  in  the  obser- 
vations themselves,  it  would  be  well,  where  great  accuracy  is  required,  to 
increase  the  number  of  these  equations,  and  treat  them  after  the  method 
of  least  squares. 

15.  —  The  finding  circles. — These  may  indicate  zenith  distances, 
altitudes,  or  polar  distances.  The  rule  for  adjusting  is  the  same  for  all. 

Direct  the  telescope  to  the  distant  horizon,  and  move  it  till  the. image 
of  some  small  object  appear  midway  between  the  double  axial  wires : 
clamp  the  axis,  move  the  index-arm  till  its  level  indicates  the  same  read- 
ing at  both  ends  of  the  bubble,  and  note  the  reading  of  the  vernier. 
Unclamj  and  reverse  the  axis ;  bring  the  image  of  the  same  object  again 
between  th;  same  wires,  and  clamp  the  axis ;  move  the  index-arm  till  the 
bubble  has  tho  same  reading  at  each  end,  and  again  note  the  reading  of  the 
vernier.  If  the  vernier  reading  be  the  same  as  before,  the  circles  are  in 
adjustment ;  if  not,  add  the  readings  together,  take  the  half  sum,  move  the 
index-arm  till  the  vernier  is  brought  to  the  reading  indicated  by  this  half 
sum,  clamp  the  index-arm,  and  bring  the  air-bubble  so  as  to  have  the  same 
reading  at  each  end  by  the  adjusting  screws  of  the  level.  It  would  be  well 
to  verify  by  repeating  the  process.  It  may  be,  that  the  finders  are  gradu- 
ated from  0°  to  360°,  in  which  case,  if  the  first  reading  were  a°,  the 
second  ought  to  be  360°  — a°. 

16. — The  adjustments  in  azimuth,  collimation,  and  level  being  per- 
fected, the  middle  normal  wire  will  be  a  visible  representation  of  that 
portion  of  the  celestial  meridian  to  which  the  telescope  is  pointed ;  and 
when  a  star  is  seen  to  cross  this  wire  in  the  telescope,  it  is  in  the  act  of 
culminating.  The  precise  instant  of  this  event  being  noted  by  the  clock 
or  chronometer,  the  time  of  meridian  passage  is  known,  and  any  error 
in  noting  this  precise  time  is  lessened  by  the  use  of  the  lateral  wires  of 
the  reticle,  as  already  explained. 

17.  —  Besides,  these  lateral  wires  increase  the  chances  of  securing  an 
observation  that  might,  without  them,  be  lost.  It  frequently  happens 
that  efforts  to  obtain  the  time  of  a  body's  passing  the  middle  or  other  wire 
are  defeated  by  the  presence  of  clouds,  or  other  accidental  circumstances, 


266 


SPHERICAL    ASTRONOMY. 


m  which,  if  the  time  of  passing  any  one  be  obtained,  that  of  passing  the 
middle  or  mean  place  of  the  wires,  when  not  equally  distant,  may  be 
Deduced  thus. 

Let  <„  £2,  £3,  &c.,  be  the  times  of  crossing  the  several  wires  in  order, 
hen  will 


(18) 


n  which  tm  denotes  the  time  of  the  body's  crossing  the  mean  position  of 
ihe  wires,  and  n  the  number  of  wires.     And 
(tm—t}).cos8=i},  ' 
t—tz .  cos  d=i 


—*. cos  <= 


(19) 


in  which  S  denotes  the  declination  of  the  body  observed,  and  z,,  i2,  ?3 . . .  ?'„, 
the  constant  intervals  of  time  required  for  a  body  in  the  equator  to  pass 
over  the  distances  which  separate  the  several  wires  from  their  mean 
position. 

Adding  equations  (19)  together,  we  obtain 

tm=  —  +  -^-r (20) 

n       n  cos  0 

in  which  2  denotes  the  algebraic  sum  of  the  quantities  expressed  by  the 
letter  written  after  it. 

By  carefully  observing  a  star  whose  declination  is  known,  we  obtain 
the  values  of  i,,  4,  &c. ;  and  these  being  tabulated  with  their  proper  signs, 
equation  (20)  will  give  the  time  of  a  body's  passing  the  mean  position 
from  the  time  of  passing  one  or  more  of  the  threads. 

The  Collimating  Telescope. 

1.  —  In  some  situations  it  would  not  be  possible  to  obtain  a  distant 
mark  by  which  to  collimate,  and  a  near  one  could  not  be  used  in  conse- 
quence of  its  image  falling  too  far  behind  the  reticle.  In  such  cases 
recourse  must  be  had  to  what  is  called  the  collimating  telescope. 

Fif.  28. 


This  is  a  telescope  whose  eye-piece  is  removed,  and  upon  its  tube  is 
mounted  a  small  swing- frame,  supporting  a  reflector,  by  means  of  which 


APPENDIX   II.  267 

sufficient  light  may  be  thrown  through  the  telescope  to  illuminate  a  pair 
of  cross  wires,  situated  at  the  solar  focus  of  the  object-glass. 

In  this  position  of  the  wires,  we  have,  from  the  principles  of  optics, 
these  facts,  viz. :  the  rays  composing  the  pencil  of  light  proceeding  from 
any  point  of  the  cross,  will  emerge  from  the  collimator  parallel  to  a  line 
drawn  through  that  point  and  the  optical  centre  of  the  lens ;  and  if  the 
telescope  of  the  transit  be  directed  towards  the  collimator  so  as  '.o  receive 
these  rays,  an  image  of  the  point  in  question  will  appear  in  its  solar  focus, 
and  on  a  line  drawn  through  the  optical  centre  of  its  object-glass,  par- 
allel to  these  same  rays. 

The  Vertical  Collimator. 

1.  —  This  instrument  is  used  for  the  double  purpose  of  collimating, 
and  for  finding  the  zenith  or  horizontal  point  of  circles,  used  in  the  meas- 
urement of  vertical  angles.      It  consists  of   a   collimating  telescope    T 
mounted  in  a   vertical   position   upon 

an  annular  plate  R,  of  cast-iron,  float- 
ing upon  the  free  surface  of  mercury, 
contained,  in  an  annular  trough  S,  also 
of  cast-iron.  The  annular  plate  is  called 
the  float.  The  telescope  is  mounted 
upon  the  float  in  a  manner  similar  to 
the  transit,  except  that  the  axis  is  near- 
er to  the  object  end.  One  of  the  Ys 
may  be  elevated  or  depressed  by  an  ad- 
justing screw  A,  while  the  telescope  is 
turned  about  its  axis  by  another  A1 ',  thus 
affording  the  means  of  giving  the  line 
joining  the  cross  wires  and  the  optical  centre  of  the  lens  a  vertical  position. 
L  is  the  lamp,  and  G  the  reflector,  to  catch  its  light  and  throw  it  upon 
the  cross  wires  at  the  lower  end  of  the  tube. 

2.  —  The  collimatiny  process. — J'ake  the  transit  for  instance.     Level 
the  axis  carefully  ;  turn  the  telescope  in  a  vertical  position ;  place  th« 
collimator  below,  and  bring  the  image  of  the  intersection  of  its  cross  wires, 
seen  upon  the  bright  ground   £r,  accurately   on  the  intersection  of  the 
middle  wires  in   the  transit,  by   means  of  the  adjusting  screws  of  the 
collimator ;  next  turn  the  float  in  azimuth  through  1 80°.     If  the  emer- 
gent rays  from  the  collimator  be  vertical,  the   image  of  the  intersection 
of  the  collimator's  wires  will  remain  stationary,  but  if  not,  the  image  will 
move  in  the  circumference  of  a  circle ;  because,  the  plane  of  floatation 


268  SPHERICAL   ASTRONOMY. 

remaining  the  same,  the  emergent  rays  from  the  collimator  will  preserve 
their  inclination  to  the  horizon  unchanged,  thus  causing  the  line  through 
the  optical  centre  of  the  transit's  lens,  and  parallel  to  these  rays,  to  de- 
scribe a  conical  surface.  The  axis  of  this  cone,  which  is  a  vertical  line, 
is  the  position  for  the  line  of  collimation.  Supposing,  then,  the  image 
to  have  changed  its  position  during  the  semi-rotation  of  the  float,  renew 
the  contact  of  the  image  and  wires ;  one  half  by  the  adjusting  screws  of 
the  collimator,  and  the  other  half  by  a  motion  of  the  transit  and  the 
adjusting  screws  of  the  diaphragm  of  its  wires.  This  process  being  re- 
peated once  or  twice,  the  adjustment  is  made. 

3.  —  The  zenith  or  horizontal  points. — Direct  the  telescope  of  any 
circle  to  the  collimator,  and  bring  the  image  of  the  intersection  of  the 
cross  wires  in  the  collimator  to  the  line  of  collimation  ;  read  the  circle, 
and  revolve  the  float  through  an  azimuth  of  180°  ;  renew  the  contact  of 
the  image  line  of  collimation  by  moving  the  circle,  if  necessary,  and  read 
again ;  denote  the  first  reading  by  a,  the  second  by  a',  and  that  of  the 
zenith  point  by  z,  and  we  have 

a-\-a' 

,=  180'+  Jt_; 

and  denoting  the  reading  of  the  horizontal  point  by  k. 


The  Collimating  Eye-piece. 

4.  —  If  now  the  swing-frame  and  its  reflector  be  transferred  from  the 
collimating  telescope  to  the  eye-piece  of  the  telescope  of  the  instrument  sup- 
posed to  be  vertical  over  a  basin  of  mercury,  this  latter  telescope  becomes 
its  own  vertical  collimator  by  reflection,  on  applying  the  lamp  to  the  swing 
reflector.  By  perforating  the  swing  reflector,  and  applying  the  eye  behind 
it,  two  sets  of  wires  will  be  seen  in  the  solar  focus  of  the  telescope,  and 
the  collimating  process  consists  in  making  the  wires  of  Fig.  25. 

one  of  these  sets  coincident  with  those  of  the  other,  by 
the  joint  motion  of  the  telescope  and  its  reticle.  The 
little  swing  reflector,  with  a  single  microscope  as  an  eye- 
piece, just  behind  its  perforation,  to  magnify  the  wires 
and  their  images,  constitutes  the  collimating  eye-piece. 
This  beautiful  little  instrument,  which  has  done  so  much 
to  facilitate  the  process  of  collimating  an  1  the  measure- 
ment of  zenith  or  nadir  distances,  is  due  to  Professor 
J3ohnenberger  of  Tubingen. 


APPENDIX   II. 


269 


The  Mural  Circle. 

1 .  —  By  means  of  the  transit  and  a  time-keeper,  distances  are  meas- 
ured on  the  equinoctial  in  time  ;  and  by  an  easy  reduction  this  time  is 
converted  into  arc.  The  object  of  the  Mural  Circle  is  to  measure  dis- 
tances on  the  meridian. 

This  instrument  consists  of  a  metallic  circle  A  A,  varying  in  diameter 
from  four  to  eight  feet,  strongly  framed  together  or  cast  in  one  entire 
piece,  and  a  telescope,  of  considerable  optical  power,  having  a  focal  length 
about  equal  to  the  diameter  of  the  circle.  The  circle  is  firmly  attached 

Fig.  26. 


to  the  larger  end  of  a  hollow  conical-shaped  axis  at  right  angles  to  its 
plane,  which  axis  is  mounted  on  Ys,  placed  in  an  opening  through  a 
heavy  wall,  whose  front  face  is  in  the  plane  of  the  meridian.  The  gradu- 
ation is  usually,  though  not  always,  upon  the  outer  rim,  and  the  readings 
are  made  by  a  pointer  and  six  or  more  reading  microscopes  F,  mounted 
upon  the  face  of  the  wall,  at  equal  distances  from  each  other,  around  the 
circle.  The  telescope  is  mounted  upon  the  front  face  of  the  circle,  so  as 


270  SPHERICAL    ASTRONOMY. 

to  move  paiallel  to  the  plane  of  the  latter  by  means  of  a  second  axis, 
which  turns  freely  and  concentrically  within  that  of  the  circle.  The  axis 
of  the  telescope  is  also  conical,  and  is  kept  in  place  and  proper  contact 
with  that  of  the  circle,  by  means  of  a  strong  nut,  which  receives  a  screw 
cut  upon  its  smaller  end,  the  head  of  the  nut  bringing  up  against  the  end 
of  the  circle's  axis.  By  turning  this  screw  in  the  direction  of  its  thread, 
the  two  axes  are  brought  as  closely  in  contact  as  may  be  found  desirable. 

Permanently  connected  with  each  end  of  the  telescope  is  a  clamping 
arrangement,  for  the  purpose  of  seizing  the  rim  of  the  circle,  and  when 
these  are  in  bearing,  the  telescope  can  only  move  with  the  circle,  and 
when  loose,  it  may  move  independently,  thus  affording  the  means  of  meas 
uring  the  same  angular  distance  on  different  parts  of  the  circle. 

Five  clamping  and  tangent  screw  arrangements  are  permanently  at- 
tached to  the  face  of  the  wall,  for  the  purpose  of  restricting  the  motion 
of  the  circle  to  the  minute  adjustments  necessary  to  complete  the  contact 
of  the  objects  observed  with  the  reticle  of  the  telescope,  and  to  secure  the 
instrument  till  the  readings  are  made  and  recorded.  They  are  made  thus 
numerous,  that  one  may  always  be  at  hand,  in  the  various  positions  of  the 
observer  about  the  circle ;  one  of  them  is  shown  at  E. 

The  proportions  of  the  whole  instrument  are  so  adjusted  as  to  throw 
its  centre  of  gravity  on  the  axis  just  behind  the  circle,  and  between  it 
and  the  wall,  where  the  axis  is  received  by  a  stirrup  with  friction-rollers 
C  (7,  the  stirrup  being  connected  by  rods  D  D  with  levers  and  counter- 
poising weights,  which  take  the  bearing  from  the  Ys. 

The  front  Y,  or  that  nearest  the  circle,  is  movable  in  azimuth  about  a 
vertical  pintle,  and  that  at  the  smaller  end  admits  of  both  a  vertical  and 
horizontal  motion,  by  means  of  two  sets  of  antagonist  screws. 

The  tube  of  the  telescope  is  perforated  on  the  side  opposite  that  of  the 
axis  to  admit  the  light  from  a  lamp  at  a  short  distance  in  front  of  the 
circle ;  this  light  is  received  upon  a  perforated  reflector  within,  after  the 
manner  of  the  transit,  and  thrown  to  the  eye  to  illuminate  the  field  of 
view  in  nocturnal  observations.  The  intensity  of  the  illumination  is  reg- 
ulated by  square  perforations  in  two  sliding  plates,  placed  over  the  aper- 
ture in  the  tube,  and  so  connected  with  rack  and  pinion  work  as  to  move 
in  opposite  directions,  on  turning  a  large  milled-headed  screw  near  the 
eye-glass ;  one  of  the  diagonals  of  each  square  being  placed  in  the  direc- 
tion of  the  motion  of  the  plates,  the  figure  of  the  opening  will  be  un- 
changed, while  its  size  may  be  varied  at  pleasure. 

At  P  and  P  are  two  small  t^es,  permanently  fixed  to  that  of  the 
telescope,  and  at  right  angles  to  its  length.  They  are  cut  away  on  one 


APPENDIX   II.  271 

side  at  the  middle,  and  each  is  closed  at  one  end  by  a  small  disk  of 
mother-of-pearl,  movable  about  an  axis  perpendicular  to  its  plane,  and 
concentric  with  the  tube.  Between  the  disk  and  middle  of  the  tube  is  a 
convex  lens,  which  admits  of  a  motion  in  the  direction  of  the  tube,  and 
by  which  an  image  of  a  small  eccentric  perforation  in  the  disk  is  formed 
about  the  middle  of  the  cut,  and  of  course  on  one  side  of  the  axis.  Aj 
motion  of  the  pearl  causes  this  image  to  describe  the  circumference  of  a 
circle,  of  which  the  centre  is  on  the  axis  of  the  tube.  In  the  opposite 
end  of  the  tube  is  a  small  microscope  to  view  this  image.  The  image  is 
technically  called  the  ghost,  being  a  visible  but  unsubstantial  representa- 
tion of  the  perforation. 

A  small  metallic  style  projects  from  the  face  of  the  wall  at  S,  from  the 
end  of  which  may  be  suspended  a  plumb-line  of  fine  silver  wire,  with  its 
bob  immersed  in  a  vessel  of  water  or  other  liquid  at  the  bottom  of  the 
wall.  The  style  is  so  arranged  by  an  adjusting  screw  as  to  bring  the 
plumb-line  to  intersect  the  axes  of  the  small  tubes  in  the  cuts,  or  to  throw 
it  clear  of  the  instrument,  at  pleasure. 

In  the  tail  end  of  the  telescope,  and  at  the  solar  focus  of  the  object- 
glass,  is  a  reticle,  of  which  the  axial  wires  are  parallel  to  the  axis  of  the 
circle.  An  additional  wire  is  driven  by  a  micrometer  screw  in  the  direc- 
tion, perpendicular  to  the  axial  wires,  while  it  is  also  kept  constantly  par- 
allel to  them. 

The  telescope  has  a  collimating  eye-piece,  which  is  used  for  the  same 
purpose  and  in  the  same  manner  as  in  the  transit. 

Adjustments. 

2.  —  The  adjustments  are,  first,  to  make  the  line  of  collimation  per- 
pendicular to  the  axis,  and,  second,  to  make  the  axis  perpendicular  to  the 
meridian.     The  plane  of  the  circle  and  tube  of  the  telescope  are  placed 
at  right  angles  to  the  axis  by  the  manufacturer ;  the  face  of  the  wall  is 
built  as  nearly  in  the  meridian  as  possible  by  the  aid  of  meridian  marks  ; 
and  the  Ys  are  so  placed  as  to  bring  the  axis,  when  mounted,  nearly  per- 
pendicular to  the  face,  so  that  the  adjustments  are  approximately  made 
when  the  instrument  is  put  up.     To  complete  them,  begin  with 

3.  —  The  line  of  collimation. — Turn  the  circle  till  the  telescope  is 
vertical,  suspend  the  plumb-line  and  bring  it  by  its  adjusting  screw  to  co- 
inci  ie  with  the  upper  ghost  as  seen  through  the  microscope :  examine 
the  position  of  the  lower  ghost ;  if  it  be  not  on  the  line,  turn  the  pearl 
about  its  axis  till  it  is :  clear  the  line  from  the  instrument,  and  invert  the 
telescope  by  revolving  the  circle  through  180°;  bring  the  line  to   the 


272  SPHERICAL   ASTRONOMY. 

upper  ghost  as  before,  and  again  examine  the  lower  ghost ;  if  it  be  on 
the  line,  the  axis  of  the  circle  is  horizontal,  but  if  not,  bring  it  to  the 
line,  one-half  by  the  vertical  adjusting  screws  of  the  circle's  axis  and  half 
by  a  revolution  of  the  pearl.  When  by  repeating  this  process  once  or 
twice  the  axis  is  made  horizontal,  put  on  the  collimating  eye-piece,  and 
directing  the  telescope  to  the  trough  of  mercury  at  the  foot  of  the  pier, 
and  immediately  below,  move  the  diaphragm  of  the  cross  wires  till  the 
wire,  which  is  perpendicular  to  the  axis,  coincides  with  its  image — the  line 
of  collimation  will  be  in  a  vertical  plane,  and  of  course  perpendicular  to 
the  axis,  which  is  horizontal. 

Should  the  telescope  have  no  collimating  eye-piece,  recourse  may  be  had 
to  the  vertical  collimator,  which  is  to  be  used  exactly  as  in  the  transit. 

Since  reflection  takes  place  in  a  plane  normal  to  the  reflecting  surface, 
the  axis  may  be  made  horizontal  by  observing  the  same  star  directly, 
and  by  reflection  from  the  free  surface  of  mercury.  If  the  time  of  the 
star's  appearing  on  the  line  of  collimation  in  both  views  be  the  same,  the 
two  positions  of  the  line  of  collimation  will  lie  m  the  same  vertical  plane, 
and  being  equally  inclined  to  the  horizon,  the  axis  with  which  they  make 
a  constant  angle  must  be  horizontal. 

4.  —  Axis  perpendicular  to  the  meridian. — This  adjustment  may  be 
made  by  the  method  pointed  out  for  the  same  adjustment  in  the  transit ; 
and  when  not  perfected,  the  amount  of  error  may  be  found  by  the  process 
explained  for  that  instrument. 

Polar  and  horizontal  points. — On  the  circumference  of  the 
circle  is  a  scale  of  equal  parts,  each  part  having  an  angular  value  of  five 
minutes.  Every  twelfth  division  is  numbered,  the  numbers  varying  from 
1  to  360°  inclusive  ;  these  indicate  the  degrees  of  the  scale;  and  to  facil- 
itate the  reading,  the  intermediate  divisions  are  also  numbered,  but  in 
smaller  characters. 

If  the  reading  be  known  when  the  line  of  collimation  is  either  hori- 
zontal or  directed  to  the  pole  of  the  heavens,  and  the  reading  be  taken 
when  directed  upon  the  centre  of  any  body  as  it  passes  the  meridian,  the 
difference  of  the  readings  will  in  the  first  case  be  the  observed  meridian 
altitude  of  the  body,  and  in  the  second  its  observed  polar  distance. 

5.  —  The  horizontal  point. — This  is  found  by  means  of  the  collima- 
ting eye-piece,  or  vertical  coll^mator,  by  the  process  indicated  at  page  268, 
or  as  follrws,  viz. :  having  carefully  ascertained  the  value  of  a  revolution 
of  the  micrometer  in  the  eye-piece  of  the  telescope,  and  the  reading  of  its 
divided  head  when  the  movable  wire  is  coincident  with  that  parallel  to 

he  axis,  set  the  telescope  nearly  in  the  position  at  which  a  star  would 


APPENDIX    II 


273 


appear  by  reflection  on  the  stationary  wire  ;  clamp  the  circle  and  record 
the  reading  of  the  index  and  microscopes ;  when  the  star  is  at  a  conve- 
nient distance  from  the  meridian  wire,  bisect  it  by  the  movable  wire  with- 
out moving  the  circle,  and  note  the  time  accurately.  Unclamp  the  circle, 
and  bring  the  star  by  direct  view  accurately  on  the  stationary  wire,  by 
turning  the  whole  circle  about  its  axis ;  again  note  the  time,  and  record 
the  reading  by  the  index  and  microscopes.  Denote  by  R  the  first  read- 
ing, by  D  the  second,  and  by  m  the  angular  value  of  the  distance  between 
the  fixed  and  movable  wire,  as  indicated  by  the  micrometer ;  then,  if  the 
star  had  been  observed  accurately  on  the  meridian,  would  the  reading  of 
the  horizontal  point  be 

R  ±  m-\-  D 


Fig.  27. 


since  the  star  must  appear  as  far  below  the  horizon  by  reflection  as  it 
actually  is  above  it.  But  as  the  star  cannot  be  taken  at  the  same  instant 
in  both  positions  of  the  instrument,  the  readings  R  and  Z>,  taken  as  above 
indicated,  must  be  reduced  to  what  they  would  have  been  if  taken  on  the 
meridian. 

6.  —  This  correction  will  now  be  explained. 
Ler.  S'  S  S"  be  the  small  diurnal  circle 
of  the  star  ;  P  M  S'  an  arc  of  the  meridian  ; 
£  tltf  position  of  the  star  when  observed 
on  the  intersection  of  the  axial  and  one  of 
the  side  normal  wires ;  MS  C  the  arc  of 
a  great  circle,  of  which  the  axial  wire  is  a 
portion.  The  point  M  will  be  that  to 
which  the  line  of  collimation  is  actually 
directed,  and  S'  is  that  in  which  the  star 
will  reach  the  meridian  ;  the  arc  M Sf  is, 
therefore,  the  reduction  to  the  meridian. 

Make       P  —  MP  S  =  hour  angle  of  star ; 

d  =  P  S  —  polar  distance  of  star ; 

y  =  P  M  =  polar  distance  of  line  of  collimation ; 

x  =  M  S'  =  reduction  to  meridian. 

Then  in  the  triangle  MP  S,  right-angled  at  M, 

sin  y  cos  d 


cos  P  =  tan  y  .  cot  d  = 


cos  y '  sin  d  ' 


and  subtracting  this  from 


1  =  1, 

18 


274  SPHERICAL    ASTRONOMY. 

we  have,  after  reducing,  and  replacing  1—  cos  P  by  2  sin8  \P, 


cos  y  .  sin  a 

Tlie  observation  being  made  very  near  the  meridian,  P  and  d—y  will  b» 
very  small,  and  hence 

2sm2  JP    =  2.(JP.sin  I")2  =  £P2  .  sin8  1"; 
sin  (d  —  y)  =  sin  #  =  rr  .  sin  1"  ; 
sin  c?  .  cos  y  =  ^  sin  2  c?,  very  nearly. 

which  in  the  above  equation  give,  after  reduction, 
a?  =  Jsin  2c?.P2.sin  1", 

in  which  P  is  expressed  in  seconds  of  arc.     To  express  it  in  time,  make 
P=15  Pb  and  we  shall  finally  have 

OOK 

x=~.nn2d.Pf.wnI", 

P,  denoting  the  number  of  seconds  of  time  in  the  hour  angle  of  the  star. 

If,  now,  the  numbers  on  the  circle  be  supposed  to  increase  in  the  direc- 
tion from  the  pole  to  the  zenith,  and  the  observed  reading  be  denoted  by 
R,  then,  since  the  line  of  collimation  is  nearer  the  pole  than  tht-  place  of 
culmination  of  the  star,  will  the  true  reading  be 

225 
R  —  x  =  R  --  -sin  2d.  P,2  sin  1"      .     .     .     .     (21) 

for  all  stars  whose  declinations  are  of  the  same  name  as  the  latitude  of 
the  place,  and  above  the  pole,  and 

OOK 

R  +  x=:R+--sm2d.P*sml"  .     ....     (221 

for  all  stars  below  the  pole,  or  whose  declinations  are  not  of  the  same 
name  as  the  latitude. 

7.  —  The  interval  P  is  obtained  from  the  indications  of  a  time- 
keeper. This  usually  runs  too  fast  or  too  slow.  To  get  the  true  from  the 
indicated  interval,  suppose  the  time-keeper  to  gain  or  lose  a  seconds  du- 
ring one  revolution  of  the  earth  upon  its  axis.  Ifenote  by  A.  the  number 
of  sidereal  seconds  in  the  time  of  this  revolution,  and  by  t  the  true  interval 
Fouht  then  will 


APPENDIX   II.  275 

A  ±  a  :  A  : :  P  :  t, 

t-      A       P-  —    P- 

"A±~afl  .a' 

A 

in  which  P  is  the  indicated  interval. 

Developing  the  coefficient  of  P,  and  limiting  the  series  to  the  first 

power  of  — ,  because  a  is  usually  a  small  number  of  seconds,  we  have 


or  replacing  A  by  its  value  86.400, 

t  —  (1  T  .  000012  a)  P  =  a  .  P, 
in  which 

a  =  1  qp  .  000012  a. 

Substituting  aP  for  P,  in  equations  (21)  and  (22),  and  making 

i  =  a2  =  1  ^:  .  000023  a, 
there  will  result  for  the  true  reading 

OOPi 

JK=F--.t.P*.sm2c*  sin  1"    .....     (23) 

8.  —  Denote  by  D  and  J2  the  readings  of  the  circle  by  the  direct 
and  reflected  views  ;  by  x  and  x'  the  corresponding  reductions  to  the  me- 
ridian ;  by  m  the  small  difference  observed  between  the  angle  of  incidence 
and  reflection,  and  by  H  the  reading  of  the  horizonta.  point  ;  then  will 


and 


_  _  m      x—x 

~~~  ~~~        2  "  ~T~ 


9.  —  Value,  in  arc,  of  units  on  the  screw-head  connected  with  the 
movable  wire.  —  Run  the  movable  wire  to  one  edge  of  the  field  of  view, 
say  the  upper,  and  bring  it  by  a  motion  of  the  circle  upon  some  well- 
defined  and  distant  object  ;  read  the  circle  and  micrometer  ;  run  the  wire 
to  the  opposite  or  lower  edge  of  the  field,  and  by  a  motion  of  the  circle 
bring  the  wire  to  same  object  again  ;  read  the  circle  and  micrometer  as 
before,  and  divide  the  difference  of  the  circle  readings,  reduced  to  seconds; 
by  the  difference  of  the  micrometer  readings,  expressed  in  units  of  the 
screw-head  ;  the  quotient  will  be  the  value  sought.  Or, 

Invert  the  telescope  over  a  basin  of  mercury,  by  moving  the  circle,  and 


27-6 


SPHERICAL    ASTRONOMY 


bring  the  image  of  the  movable  wire,  supposed  at  one  edge  of  the  field, 
to  coincide  with  the  wire  itself;  read  the  circle  and  micrometer:  move 
the  wire  to  the  opposite  edge,  and  turn  the  circle  till  the  wire  and  its 
image  again  coincide,  and  read  as  before ;  divide  the  difference  of  the 
circle  readings,  reduced  to  seconds,  by  the  difference  of  the  micrometer 
readings  expressed  in  units  of  its  screw-head ;  the  quotient  will  be  the 
value  sought. 

Altitude  and  Azimuth  Instrument. 

\ .  —  This  instrument,  as  its  name  indicates,  is  employed  in  the 
measurement  of  vertical  and  horizontal  angles.  It  has  two  graduated 
circles  and  a  telescope.  The  planes  of  the  circles  are  at  right  angles  to 
each  other ;  one  called  the  azimuth  circle,  being  connected  with  a  tripod, 
by  which  it  is  levelled  and  kept  in  a  horizontal  position ;  while  the  other, 
called  the  altitude  circle,  is  mounted  upon  a  horizontal  axis,  with  which 
the  telescope  is  also  united,  after  the  manner  of  the  transit. 

To  the  centre  of  the  tripod 
A  A  is  fixed  a  vertical  axis,  of 
a  length  equal  to  about  the 
radius  of  the  circle ;  it  is  con- 
cealed from  view  by  an  exterior 
cone  B.  On  the  lower  part  of 
the  axis,  and  in  close  contact 
with  the  tripod,  is  centred  the 
azimuth  circle  (7,  which  admits 
of  a  horizontal  circular  motion 
of  about  three  degrees,  for  the  ' 
purpose  of  bringing  its  zero  ex- 
actly in  the  meridian;  this  is 
effected  by  a  slow  moving- 
icrew,  the  milled  head  of  which 
is  shown  at  D.  This  motion 
should,  however,  be  omitted  in 
instruments  destined  for  exact 
work,  as  the  bringing  the  zero 
into  the  meridian  is  not  requi- 
site, either  in  astronomy  or  sur- 
veying :  it  is,  in  fact,  purchasing  A 
a  convenience  too  dearly,  by 
introducing  a  source  of  error 


APPENDIX    II.  277 

not  always  trivial.  Above  the  azimuth  circle,  and  concentric  with  it,  is 
placed  a  strong  circular  plate  E,  which  carries  the  whole  of  the  upper 
works,  and  also  a  pointer,  to  show  the  degree  and  nearest  five  minutes  to 
be  read  off  on  the  azimuth  circle  ;  the  remaining  minutes  and  seconds 
being  obtained  by  means  of  the  two  reading  microscopes  F.  This  plate, 
by  means  of  the  cone  B,  rests  on  the  axis,  and  moves  concentrically  with 
it.  The  conical  pillars  H  support  the  horizontal  or  transit  axis  /,  which, 
being  longer  than  the  distance  between  the  centres  of  the  pillars,  the  pro- 
jecting pieces  c,  fixed  to  their  top,  carry  out  the  Ys  a,  to  the  proper  dis- 
tance, for  the  reception  of  the  pivots  of  the  axis ;  the  Ys  are  capable  of 
being  raised  or  lowered  in  their  sockets  by  means  of  the  milled-headed 
screws  6,  for  a  purpose  hereafter  to  be  explained*  The  axis,  with  its  load; 
is  prevented  from  pressing  too  heavily  on  its  bearings,  by  two  friction* 
rollers,  on  which  it  rests ;  one  of  these  rollers  is  shown  at  e.  A  spiral 
spring,  fixed  in  the  body  of  each  pillar,  presses  the  rollers  upward,  with  a 
force  nearly  a  counterpoise  to  the  superincumbent  weight ;  the  rollers  on 
receiving  the  axis  yield  to  the  pressure,  and  allow  the  pivots  to  find  their 
proper  bearings  in  the  Ys,  relieving  them,  however,  from  a  great  portion 
of  the  weight. 

The  telescope  K  is  connected  with  the  horizontal  axis,  as  before  re- 
marked, in  a  manner  similar  to  that  of  the  transit  instrument.  Upon  the 
axis,  as  a  centre,  and  in  contact  with  the  telescope  on  either  side,  is  fixed 
the  double  circle  J.  The  circles  are  united  by  small  brass  bars  ;  by  this  cir- 
cle the  vertical  angles  are  measured,  and  the  graduations  are  cut  on  a  narrow 
ring  of  silver,  inlaid  on  one  of  the  sides,  which  is  usually  termed  the  face  of 
the  instrument :  a  distinction  essential  in  making  observations.  The  clamp 
for  fixing,  and  the  tangent-screw  for  giving  a  slow  motion  to  the  vertical 
circle,  are  placed  beneath  it,  between  the  pillars  H,  and  attached  to 
them,  as  shown  at  L.  A  similar  contrivance  for  the  azimuth  circle 
is  represented  at  M.  The  reading  microscopes  for  the  vertical  circle 
are  supported  by  two  arms  bent  upward  near  their  extremities,  and 
attached  to  one  of  the  pillars.  The  projecting  arms  are  shown  at  N 
and  the  microscopes  above  at  0,  the  latter  admitting  of  a  slight  motion 
by  means  of  antagonistic  adjusting  screws  independently  of  the  sup- 
porting arms. 

A  reticle  consisting  of  five  equidistant  axial  and  as  many  equidistant 
normal  wires,  is  in  the  principal  focus  of  the  object-glass.  The  illumina- 
tion of  the  wires  at  night  is  by  a  lamp,  supported  near  the  top  of  one  of 
the  pillars  at  rf,  opposite  the  end  of  one  of  the  pivots  of  the  axis,  which, 
l»eino  perforated,  admits  the  light  to  the  centre  of  the  telescope  tube, 


278  SPHERICAL  ASTRONOMY. 

where,  falling  on  a  diagonal  reflector,  it  is  reflected  to  the  eye,  and  illu 
mines  the  field  of  view. 

The  vertical  circle  is  usually  divided  into  four  quaarants,  each  num- 
bered 1°,  2°,  3°,  &c.;  up  to  90°,  and  following  one  another  in  the  same 
order  of  succession  ;  consequently,  in  one  position  of  the  instrument  alti- 
tudes are  read  off,  and  with  the  face  of  the  instrument  reversed,  zenith 
distances ;  and  an  observation  is  not  to  be  considered  complete  till  the 
object  has  been  observed  in  both  positions.  The  sum  of  the  two  readings 
will  always  be  90°,  if  there  be  no  error  in  the  adjustments,  in  the  circle 
itself,  or  in  the  observations. 

•  It  is  necessary  that  the  microscopes  0  and  the  centre  of  the  circle 
should  occupy  the  line  of  its  horizontal  diameter ;  to  effect  which,  an  up- 
and-down  motion,  by  means  of  the  screws  6,  is  given  to  the  Ys.  A 
'spirit-level  P  is  suspended  from  the  aims  which  carry  the  microscopes : 
this  shows  when  the  vertical  axis  is  set  perpendicular  to  the  horizon.  A 
scale,  usually  showing  seconds,  is  placed  along  the  glass  tube  of  the  level, 
which  exhibits  the  amount,  if  any,  of  the  inclination  of  the  vertical  axis. 
This  should  be  noticed  repeatedly  whilst  making  a  series  of  observations, 
to  ascertain  if  any  change  has  taken  place  in  the  position  of  the  instru- 
ment after  its  adjustments  have  been  completed.  One  of  the  points  of 
suspension  of  the  level  is  movable,  up  or  down,  by  means  of  the  screw  f, 
fo  the  purpose  of  adjusting  the  bubble.  A  striding-level,  similar  to  the 
one  employed  for  the  transit  instrument,  and  used  for  a  like  purpose,  restr 
upon  the  pivots  of  the  axis.  It  must  be  carefully  passed  between  the 
radial  bars  of  the  vertical  circle  to  set  it  up  in  its  place,  and  must  be  re- 
moved as  soon  as  the  operation  of  levelling  the  horizontal  axis  is  per- 
formed. The  whole  instrument  stands  upon  three  foot-screws,  placed  at 
the  extremities  of  the  three  branches  which  form  the  tripod,  and  brass 
cups  are  placed  under  the  spherical  ends  of  the  foot-screws.  A  stone 
pedestal,  set  perfectly  steady,  is  the  best  support  for  this  as  well  as  the 
portable  transit  instrument. 

Adjustments. 

2.  —  These  have  for  their  object  to  make,  1st,  the  azimathal  axis  per- 
pendicular to  the  horizon ;  2d,  to  make  the  axis  of  the  vertical  circle 
horizontal ;  3d,  to  place  the  vertical  circle  at  such  a  height  that  its  mi- 
croscopes shall  point  to  the  opposite  extremities  of  a  horizontal  diameter ; 
4th,  to  make  the  line  of  collimatiori  perpendicular  to  the  axis  of  the  alti- 
tude circle,  and  horizontal  when  the  reading  of  the  vertical  circle  is  zero. 

3.  —  The  vertical  axis. — Turn  the  instrument  about,  until  the  spirit- 


APPENDIX   11.  279 

level  P  is  lengthwise  in  the  direction  of  two  of  the  foot-screws,  when  by 
their  motion  the  spirit-bubble  must  be  brought  to  occupy  the  middle  of 
the  glass  tube,  which  will  be  shown  by  the  divisions  on  the  scale  attached 
to  the  level.  Having  done  this,  turn  the  instrument  half  round  in  azi- 
muth, and  if  the  axis  is  truly  vertical,  the  bubble  will  again  settle  in  the 
middle  of  the  tube ;  but  if  not,  the  amount  of  deviation  will  show  double 
the  quantity  by  which  the  axis  deviates  from  the  vertical  in  the  direction 
of  the  level ;  this  error  must  be  corrected,  one-half  by  means  of  the  two 
foot-screws,  and  the  other  half  by  raising  or  lowering  the  spirit-level  itself, 
which  is  done  by  the  screw  represented  at/.  The  above  process  of  rever- 
sion and  levelling  should  be  repeated,  to  ascertain  if  the  adjustment  has 
been  correctly  performed. 

Next  turn  the  instrument  round  in  azimuth  a  quarter  of  a  circle,  so  that 
the  level  P  shall  be  at  right  angles  to  its  former  position ;  it  will  then  be 
over  the  third  foot-screw,  which  may  be  turned  until  the  air-bubble  is 
again  central,  if  not  already  so,  and  this  adjustment  will  be  completed ;  if 
delicately  performed,  the  air-bubble  will  steadily  remain  in  the  middle  of 
the  level  during  an  entire  revolution  of  the  instrument  in  azimuth.  These 
adjustments  should  be  first  performed  approximately,  for  if  the  third  foot- 
screw  is  much  out  of  the  level,  it  will  be  impossible  to  get  the  other  two 
right.  The  vertical  axis  is  now  adjusted. 

4.  —  The  axis  of  the  vertical  circle. — This  adjustment  is  performed 
exactly  as  in  the  transit,  by  means  of  the  striding-level. 

5.  —  Height  of  the  vertical  circle.— The  last  adjustment  being  made, 
bring  the  microscopes  to  their  zeros,  and  turn  the  vertical  circle  slightly, 
the  striding-level  being  still  mounted,  till  some  one  of  its  divisions  be 
brought  to  the  cross  wires  of  one  of  the  microscopes.     Examine  the  other 
microscope,  and  if  its  cross  be  not  on  or  near  the  division  of  the  circle, 
1 80°  distant  from  the  first,  depress  or  elevate  the  circle  by  the  milled 
screws  b  till  it  is,  keeping  the  axis  horizontal  by  means  of  the  level ;  this 
will  give  a  sufficient  approximation  to  bring  the  error  of  adjustment  within 
the  range  of  the  adjusting  screws  which  move  the  microscopes  indepen- 
dently of  their  supporting  arm.     Recourse  must  now  be   had   to  these 
screws,  by  turning  which  in  the  direction  indicated  by  the  relative  posi- 
tion of  tha  circle  division  in  question  and  the  cross  wires,  the  adjustment  is 
perfected. 

6.  —  The  line  of  collimation. 

As  the  vertical  circle  is  not,  like  the  mural,  generally  used  as  a  differen- 
tial instrument,  but  in  the  measurement  of  absolute  altitudes  or  zenith 
distances,  it  is  not  only  necessary  that  the  line  of  collimation  shall  be  per* 


280  SPHERICAL    ASTRONOMY. 

pendieular  to  the  transit  axis,  but  also  that  it  shall  be  parallel  to  the 
radius  of  the  graduated  circle  drawn  to  the  zero  of  its  scale. 

Let  x  denote  the  angle  made  by  the  line  of  collimation  with  the  plane 
normal  to  the  transit  axis,  which  angle  is  usually  very  small,  and  a  the 
reading  of  the  azimuth  circle,  when  the  -telescope  is  pointed  to  some  well- 
defined  object  in  or  near  the  horizon.  If  the  line  of  collimation  lie  on  the 
side  of  the  normal  plane,  towards  the  zero  of  the  circle,  the  true  reading 
will  be  sensibly  equal  to 

a  —  x, 

if  there  be  no  other  error  of  adjustment. 

Now  revolve  the  instrument  in  azimuth  180°,  bring  the  telescope  again 
on  the  object,  and  denote  by  a!  the  new  reading  ;  the  tine  reading  now 
will  be 

«'+  v, 

the  difference  of  these  true  readings  is  obviously  a  semi-circumference, 

whence 

a  _  a'—  <2X  =  180°; 
and 

a—  a'-  180° 

*=-    "    -' 

and  the  true  reading  in  the  second  position  becomes 

a  -a'—\  80° 


Again,  denote  by  y  the  small  angle  which  the  line  of  collimation  makes 
with  the  plane  passing  through  the  axis  of  the  vertical  circle  and  that  zero 
of  this  latter  circle  nearest  the  line  of  collimation,  and  suppose  the  line  of 
collimation  to  lie  above  this  plane  when  the  telescope  is  directed  to  the 
same  object,  as  before.  •  Let  b  denote  the  apparent  altitude,  supposing 
the  circle  in  the  position  to  mark  altitudes  ;  the  true  altitude  is  sensibly 
equal  to 

6-hy; 

turn  the  instrument  in  azimuth  180°,  and  bring  the  telescope  again  on  the 
object  ;  the  line  of  collimation  will  now  be  below  the  plane  of  the  axis 
and  zero,  but  the  circle  now  indicates  a  zenith  distance  &',  whence  the 
true  zenith  distance  is 


adding  these  measures  together,  we  have 


APPENDIX    II  281 

&-f6'+2y  =  90° 
•  90° -(6 +6') 

y  =  -   —        » 

and  the  true  zenith  distance  becomes 

„  ,   W°-(b  +  b>) 
~2~ 

Whence  to  adjust  the  line  of  collimation  we  have  this  rule,  viz.  : 

Direct  the  telescope  to  some  well-defined  and  distant  object,  not  far  from 
the  horizon,  and  bring  its  image  to  the  intersection  of  the  middle  wires ; 
record  the  reading  of  the  azimuth  and  vertical  circles ;  turn  the  instru- 
ment in  azimuth  180°,  bring  the  line  of  collimation  again  on  the  object, 
and  record  the  new  readings  of  the  circles ;  subtract  from  the  difference 
of  the  azirnuthal  readings  180°,  divide  by  2,  and  add  (algebraically)  the 
quotient  to  the  last  azimuthal  reading  for  a  new  reading  in  azimuth.  Add 
the  two  readings  of  the  vertical  circle  together,  subtract  the  sum  from  90°, 
and  add  half  the  difference  to  the  last  reading  for  a  new  reading  on  the 
vertical  circle.  Set  the  circles  to  these  new  readings,  clamp,  and  by  the 
adjusting  screws  of  the  reticle  bring  the  line  of  collimation  to  the  object, 
and  the  adjustment  is  made ;  it  should  be  verified,  however,  by  repetition. 

7.  —  To  make  the  normal  wires  perpendicular  to  the  transit  axis, 
proceed  as  in  the  case  01  the  transit  instrument,  viz. :  Move  the  diaphragm 
about  in  its  own  plane,  till  the  image  of  some  object  appears  to  run  accti 
rately  along  some  one  of  the  wires,  say  the  middle  one,  while  the  tele- 
scope is  turned  about  its  axis. 

8.  —  The  altitude  and  azimuth  instrument  is  regarded  by  many  as 
the  most  universally  useful  of  all  astronomical  instruments.     It  is  portable 
and  accurate.     When  used  in  the  meridian,  it  may  perform  the  work  of 
the  transit  and  mural  circle,  though  with  somewhat  diminished  accuracy. 
But  its  principal  merit  consists  in  the  ease  with  which  it  may  be  moved  in 
azimuth  without  impairing  its  measurement  of  altitudes  and  zenith  dis- 
tances. 

9.  —  The  instrumental  bearing  of  an  object  is  the  angle  indicated 
by  the  reading  of  the  azimuth  circle  when  th'e  centre  of  the  object  is  ap- 
parently on  the  line  of  collimation.      From  the  instrumental,  the  true 
bearing,  or  true  meridian,  is  found  by  a  process  to  be  explained  hereafter. 

10.  —  To  find  the  altitude  and  instrumental  bearing  of  an  object  at 
any  instant,  it  is  only  necessary  to  make  the  object  pass  the  line  of  colli- 
mation by  turning  both  tangent-screws  as  it  moves  through  the  field  of 
vif»w,  and  to  note  the  time  of  passage,  ana  v<  ad  the  circles. 


282  SPHERICAL   ASTRONOMY. 

11.  —  The  altitude  and  time,  or  the  instrumental  bearing  and  time, 
are  the  elements  more  commonly  observed  in  the  case  of  celestial  objects. 

12.  —  To  obtain  the  altitude  and  time. — With  the  circles  undamped, 
direct  the  telescope,  which  it  will  be  remembered  inverts,  so  as  to  bring 
the  image  of  the  object  in  the  lower  or  upper  part  of  the  field  of  view,  as 
the  body  may  be  rising  or  setting ;  clamp  the  circles,  and  by  the  tangent 
screw  of  the  azimuth  motion,  bring  the  image  to  the  middle  normal  wire, 
and  keep  it  there  till  it  passes  all  the  axial  wires,  carefully  noting  the 
time  of  its  passing  each,  and  also  noting  the  indications  of  the  level  be- 
fore it  passes  the  first  and  after  it  passes  the  last  one.     Now  read  the 
vertical  limb,  unclamp,  and,  by  an  azimuthal  motion,  reverse  the  face  of 
the  vertical  circle  without  unnecessary  loss  of  time,  and  go  through  the 
same  operation  as  before.     Reduce  the  vertical  readings  to  the  same  de- 
nomination of  altitude  or  zenith  distance,  correct  them  by  applying  the 
level  readings,  and  take  half  the  sum  for  the  altitude  or  zenith  distance, 
as  the  case  may  be.     Add  the  times  together,  and  divide  the  sum  by  the 
number  of  recorded  times  for  the  corresponding  time. 

13.  —  To  find  the  instrumental  bearing  and  time,  direct  the  telescope 
as  before,  and  clamp ;  with  the  tangent-screw  of  the  vertical  motion,  bring 
the  image  of  the  object  to  the  middle  axial  wire,  and  keep  it  there  till  it 
passes  all  the  normal  wires,  on  each  of  which  record  the  time.     The  read- 
ing of  the  azimuth  circle  will  give  the  instrumental  bearing,  and  a  mean 
of  all  the  times  will  give  the  corresponding  time. 

All  of  this  supposes  that  the  object's  change  in  altitude  and  azimuth  is 
uniform ;  and  although  this  is  not  strictly  true,  it  is  nevertheless  so  nearly 
so  for  the  short  time  its  image  is  in  the  field  of  view,  that  the  error  will 
be  inappreciable  during  the  interval  required  for  a  single  set  of  ob- 
servations. 

The  Equatorial. 

1.  —  The  object  of  the  equatorial  or  parallactic,  as  it  is  frequently 
called,  is  to  support  a  telescope,  generally  of  great  size  and  optical  power, 
in  su;h  manner  as  to  give  to  the  observer  the  means  of  directing  it  with 
ease  io  any  part  of  the  heavens,  and  to  measure  at  once  the  apparent  hour 
angle  and  polar  distance  of  a  heavenly  body.  In  the  principles  of  its  con- 
struction, it  is  like  the  altitude  and  azimuth  instrument,  but  differs  from  it 
in  the  position  of  its  axes,  which,  instead  of  being  vertical  and  horizontal, 
are,  when  in  position,  respectively  perpendicular,  and  parallel  to  the  plane 
of  the  equinoctial.  The  first  is  called  the  polar,  the  second  the  declina- 
tion axis.  It  has  two  graduated  circles,  one  securely  attached  to  each 


APPENDIX  Ii.  283 

axis ;  the  plane  of  one,  viz.,  that  attached  to  the  polar  axis,  is  parallel,  ant) 
the  other  perpendicular  to  the  equinoctial.  The  first  is  called  the  Aowr, 
and  the  second  the  decimation  circle.  By  a  motion  of  the  polar  axis,  to 
which  the  supports  of  the  declination  axis  are  attached,  the  declination 
circle  may  be  made  parallel  to  any  assumed  declination  circle  of  the  celes- 
tial sphere.  The  polar  axis,  always  much  loaded,  is,  in  low  latitudes,  con- 
siderably inclined  to  the  horizon,  and  the  practical  difficulty  of  supporting 
it  has  given  rise  to  a  variety  in  the  form  of  the  instrument.  That  repre- 
sented in  the  figure  is  the  one  now  most  generally  used,  and  it  is  intro- 
duced here  on  that  account.  The  principle  is  the  same  in  all. 

The  supporting-stand  is  shown  at  H,  H,  H.  It  is  made  either  of  a 
strong  frame  of  wood-work,  or  is  cut  from  a  solid  block  of  stone.  B  is  a 
plate  of  metal,  firmly  secured  to  the  stand,  the  surface  of  contact  being 
parallel  to  the  axis  of  the  heavens.  Upon  this  plate  the  instrument  is 
mounted.  The  polar  axis  is  seen  at  /.  It  is  of  steel,  and  revolves  in  two 
cylindrical  collars  near  the  extremities,  and  the  lower  end,  being  rounded 
off  and  highly  polished,  rests  upon  a  steel  plate  attached  to  a  bearing- 
piece  K. 

To  the  lower  end  of  this  axis  is  attached  the  hour-circle  72,  which  is 
either  graduated  into  hours,  minutes,  and  seconds,  or  into  degrees  and  the 
usual  subdivisions,  at  the  option  of  the  person  ordering  the  instrument. 
The  verniers,  or  reading  microscopes,  and  tangent-screw  arrangement,  are 
supported  by  pieces  connected  with  the  plate  B.  Tho  declination  axis 
revolves  in  a  metallic  tube  M,  which  forms  a  part  of  the  frame-work  se- 
cured to  the  top  end  of  the  polar  axis.  To  one  end  of  the  declination  axis 
is  attached  the  declination  circle  P,  which  is  graduated  so  as  to  read  polar 
distances  or  declinations — suppose  the  former,  it»  micrometers  and  tangent- 
screw  being  mounted  upon  pieces  projecting  from  the  extremity  of  the 
tube  M,  and  to  the  other  end,  which  projects  slightly  beyond  the  frame- 
work, is  attached  the  telescope  at  a  point  nearer  the  eye-end  than  the 
middle.  The  excess  of  weight  towards  the  object-end  is,  in  the  mounting 
by  Mr.  Henry  Fitz,  of  New  York,  compensated  by  a  counterpoise  cylin- 
drical lever  within  the  tube  of  the  telescope,  and  so  arranged  in  bearing 
as  to  counteract  all  tendency  in  the  tube  to  bend.  Attached  to  the  end 
of  the  declination  axis,  is  a  counterpoise  weight  0,  the  office  of  which  is  to 
throw  the  centre  of  gravity  of  the  entire  movable  part  of  the  instrument 
in  the  polar  axis  near  its  upper  end,  where  it  is  received  by  a  pair  of  fric- 
tion-rollers. 

At  C  is  a  box  containing  a  system  of  wheel-work,  so  connected  with 
the  polar  axis  as,  by  the  lid  of  weights  and  a  centrifugal  governor,  to  give 


281 


SPHERICAL    ASTRONOMY 
Fig.  29. 


it  a  uniform  motion  of  rotation.  The  velocity  of  rotation  is  regulated  by 
a  vertical  motion  of  the  axis  of  the  governor,  whose  balls  in  their  retro- 
cession and  increasing  velocity,  force  a  pair  of  rubbing  surfaces  against  the 
interior  of  an  inverted  conical  box  :  the  moment  of  the  friction  thence 
arising  equilibrates  that  of  a  descending  weight,  and  the  motion  become* 


APPENDIX    II. 

uniform.  By  elevating  the  axis  of  the  governor,  the  motion  is  acceler- 
ated ;  by  depressing  the  axis,  it  is  retarded,  and  thus  the  velocity  of  rota- 
tion may  be  made  equal  to  that  of  the  earth  about  its  axis,  in  which  case 
a  star  in  the  field  of  view  will  be  kept  there  ly  the  instrument  itself,  the 
effect  being  the  same,  abating  refraction,  as  though  the  earth  were  at 
rest, 

2.  —  With  a  divided  object-glass  for  the  telescope,  to  be  explained 
presently,  or  with  the  position  micrometer,  the  equatorial  is  mostly  used 
as  a  differential  instrument,  and  particularly  when  the  observer  is  pro- 
vided with  a  very  full  and  accurate  catalogue  and  map  of  the  stars,  which 
serve  as  points  of  reference.     Whenever  it  is  possible  to  bring  a  known 
object  into  the  field  of  view  with  one  that  is  not  known,  the  place  of  the 
latter  is  found  by  measuring  its  bearing  and  distance  from  the  known 
object. 

3.  —  To  measure  directly  the  hour  angle  and  polar  distance  of  an 
object  with  the  equatorial,  requires  the  parts  of  the  instrument  to  be  in 
perfect  adjustment  among  one  another,  and  its  polar  axis  to  be  parallel 
to  the  axis  of  the  earth.     For  these  adjustments  and  a  full  analysis  of  the 
equatorial. 

Analysis  of  the  Equatorial. 

The  true  instrumental  position  of  an  object  is  that  indicated  by  an  in- 
strument in  perfect  adjustment  within  itself.  The  apparent  instrumental 
position  is  that  actually  indicated  by  an  instrument  whether  in  adjustment 
or  not.  When  the  several  parts  of  an  instrument  are  adjusted  with  respect 
to  each  other,  these  two  positions  are  the  same. 

The  instrumental  hour  angle  of  an  object,  is  its  angular  distance  from  a 
vertical  plane  passing  through  the  polar  axis,  estimated  upon  the  hour 
circle. 

Its  instrumental  declination  is  its  angular  distance  from  a  plane  perpen- 
dicular to  the  polar  axis,  estimated  upon  the  declination  circle;  and  its  in- 
strumental polar  distance,  its  angular  distance  from  the  polar  axis. 

The  line  of  collimation  should  be  perpendicular  to  the  declination  axis, 
and  the  latter  perpendicular  to  the  polar  axis.  The  index  of  the  hour  cir- 
cle should  stand  at  the  zero  of  the  scale  when  the  line  of  collimation  is 
parallel  to  the  vertical  plane  of  the  polar  axis,  and,  supposing  the  instru- 
ment to  read  polar  distances,  the  index  of  the  declination  circle  should  be 
at  the  zero  of  its  scale,  when  the  line  of  collimation  is  parallel  to  the  polar 
IDA, 


286  SPHERICAL    ASTRONOMY. 

Supposing  none  of  the  conditions  to  be  fulfilled,  the  apparent  instru- 
mental position  of  an  object  will  differ  from  the  true,  and  the  first  thing 
to  be  done  is  to  find  the  latter  from  the  former,  when  the  error  in  each 
of  the  above  particulars  is  known.  To  do  this,  we  will  premise  that  the 
equatorial  may  be  regarded  as  an  universal  transit  instrument,  whose 
horizon  is  the  equinoctial,  and  zenith  the  pole.  The  formulae  of  reduc- 
tion applicable  to  the  transit  will  apply  at  once  to  the  equatorial  by 
making  therein  the  symbol  for  the  latitude  90°;  in  which  case  we  shall 
have  for  the  difference  between  the  true  and  apparent  instrumental  hour 
angle  in  arc,  the  sum  of  the  last  three  terms  of  Eq.  (15),  viz., 

c  cos  (X  —  #)  sin  (X  —  §} 

-    — |—   £    %  - —         ^  ~\      &    •  ^  « 

COS  0  COS  0  COS  0 

which  reduces,  by  making  X  =  90°,  and  replacing  S  by  90°  -  -  tf,  to 
c  .  cosec  *  +  I  •  cot  <if  +  z ; 

in  which  c  is  the  error  in  the  line  of  collimation,  I  that  of  the  declination 
axis,  and  z  a  constant  correction  to  be  applied  to  every  reading  of  the  hour 
circle  arising  from  the  improper  position  of  its  index,  and  therefore  the  in- 
dex error  of  the  hour  circle,  and  *  the  instrumental  polar  distance  of  an 
object  whose  image  is  on  the  line  of  collimation. 

Denoting  by  tf '  the  true,  and  by  tf  the  apparent  instrumental  hour  angle, 
and  writing  4  <f  for  z,  we  have 

<rr  =  <f  +  dff-\-l.  cot  <ir  +  c  .  cosec  if     .     .     .     .     (a) 

Denoting  by  <ir'  the  true  instrumental  polar  distance,  and  by  4  it  the  in- 
dex error,  then  will 

*'  =  «  +  4« (6) 

Let  us  next  find  from  the  hour  angle  and  polar  distance  as  given  by  the 
instrument,  whose  parts  are  in  perfect  adjustment  among  themselves,  the 
true  hour  angle  and  polar  distance,  re- 
ferred  to  the  true  meridian  and  pole  of 
the  celestial  sphere.  It  is  obvious  that 
the  instrumental  and  true  co-ordinates 
would  not  differ,  if  the  polar  axis  of  the 
instrument  were  parallel  to  that  of  the 
heavens.  Suppose  this  latter  condition 
not  fulfilled,  and  that  the  inclination  of 
these  axes  is  very  small,  as  it  always  is, 
after  putting  the  instrument  in  position 


APPENDIX    II.  287 

after  the  manner  to  be  explained  presently.  Let  P  M  be  the  arc  of  the 
meridian  ;  P,  the  true  pole  of  the  heavens  ;  P',  the  point  in  which  the 
polar  axis  of  the  instrument  produced  pierces  the  celestial  sphere  ;  and  £, 
the  position  of  a  star.  Make  • 

p  =  P  S      =  the  true  polar  distance  ; 
•K'  =  P'  S     =  the  instrumental  polar  distance  ; 
s  =.  MP  S  =  the  true  hour  angle  of  the  star. 

The  difference  between  the  true  and  instrumental  hour  angles  will  be 
sensibly  equal  to  the  difference  of  the  angles  which  the  true  and  instru- 
mental declination  circles  of  the  object  make  with  the  plane  of  the  celestial 
and  instrumental  axes,  or  SPft—  £P'J2;  PR  being  P'  P  produced. 
Make 


9  =  M  PP',  positive  when  this  angle  is  to  the  west  of  the  meridian 


c= 


Then  in  the  triangle  SPPf,  we  have,  Napier's  Analogies, 

to*  l(P  +  !»)  =  <*&  id.  "»*<*-*) 

cos  i  (p  +  «') 

Put  P  =  180°  —  c,  and  replacing  cot  ^  d  by  its  value  in  terms  of  sin  and 
cos,  we  have 


taking  the  reciprocal,  and  observing  that  the  angles  J  (c  —  P'),  ^  d,  and 
J  (p  —  if')  are  very  small,  and  that^?  =  «•',  we  have  very  nearly 

c  —  P'  =  s  —  d'  =  d  .  cos  *'    .     .    .     ,     .     .     (c) 
But  from  the  same  triangle  we  have 

•     j       j  sin  P 

si  n  d  =  a  =  X  .  -  -  -.  ; 
sin  *      • 

and  this  in  Eq.  (c),  observing  that  the  angle  P  =  s  —  9,  gives 
s  —  <ff  =  X  .  sin  (*  —  <p)  .  cot  *'  ; 

transposing  and  replacing  tf'  by  its  value  in  Eq.  (a),  we  have,  since  s  and 
tf  only  differ  by  a  small  quantity, 

s  =  d  +  -4*  +  X  sin  (tf  —  £)  cot  «r  -f  c  .  cosec  «r  +  J  cot  *  .   .   (rf) 
With  S  as  a  pole,  and  radius  SP',  describe  the  arc  P'  T7,  then  will 


288  SPHERICAL    ASTRONOMY. 

PS=p  =  «f  +  PT. 
But  within  the  limits  supposed 

P  T  «=  X  .  cos  (s  —  <p)  =  X  .  cos  (rf  —  9)  ; 
whence,  replacing  *'  by  its  value  in  Eq.  (6),  we  have 

p  =  if  +  At  -f  X  .  cos  (<f  —  9)  .     ,     .     .     .     .     (e) 


The  adjustments  of  the  equatorial  are  of  two  classes,  viz.:  those  which 
relate  to  the  parts  among  one  another;  and  those  which  determine  the 
position  of  the  instrument  in  relation  to  the  celestial  sphere. 

The  rules  for  the  first  are  suggested  by  equations  (a)  and  (6),  and  are 
as  follows : 

Index  Error  of  the  Declination  Circle. — Direct  the  line  of  colhmation 
to  any  well-defined  object  in  any  part  of  the  horizon  ;  in  reversed  positions 
of  the  declination  circle,  the  readings  of  this  circle  in  Eq.  (I)  give 


Taking  the 'second  from  the  first, 


Apply  this  with  its  proper  sign  to  the  last  reading  Vy,  and  the  telescope 
still  being  upon  the  object,  move  the  verniers  or  microscopes  till  they  indi- 
cate this  corrected  reading. 

Line  of  Collimation. — The  preceding  correction  being  applied,  move 
the  telescope  till  the  declination  circle  marks  a  polar  distance  equal  to 
90° ;  then  by  a  motion  of  the  polar  axis,  bring  the  line  of  collimation  upon 
some  object  directly  in  the  instrumental  east  or  west ;  read  the  hour  cir- 
cle ;  reverse  the  declination  circle ;  bring  the  telescope  upon  the  same 
object,  and  read  again ;  and  these  readings,  in  Eq.  (a),  will  give,  since 

*  =  90°, 

d'  =  <f  +  4  <f  +  c, 

tf'  =  tf,—  12h  -f^tf  — c; 
whence,  subtracting  the  second  from  the  first, 

rf  -  rf,  +  12h 
2 

Apply  this  to  the  last  rending  tf,,  and  move  the  instrument  about  its  polar 


APPENDIX    II.  289 

axjs  till  the  vernier  indicates  this  reading;  then  by  a  motion  of  the  adjust- 
ing screws  which  act  upon  the  telescope,  bring  the  line  of  collimation  tc 
bear  upon  the  object. 

Declination  Axis. — Turn  the  line  of  collimation  to  an  object  directly  in 
the  instrumental  north  or  south,  to  get  the  greatest  declination.  This  will 
give  to  /  its  greatest  effect.  Read  the  hour  circle  as  before  in  the  direct 
and  reversed  position  of  the  declination  circle.  Then,  since  by  the  last 
adjustment  c  =  0,  we  have 

tf'  ==  tf  +  4  <f  +  I .  cot  «r, 

*'  =  tf,  —  12h  +  4  tf  —  /  cot  «• ; 

whence,  by  subtraction  and  reduction, 

*-*,+  12h 
/=- '- .tan*; 

or,  if  the  telescope  be  set  to  a  polar  distance  equal  to  45°, 

tf  -  tf/  +  12* 
~2~ 

Set  the  hour  circle  to  the  last  reading  tf,,  corrected  by  the  above  valiio 
of  /,  and  bring  the  line  of  collimation  back  to  the  object  by  the  adjusting 
screws,  which  act  upon  the  declination  axis. 

The  Polar  Axis  parallel  to  the  Axis  of  the  Heavens. — About  the  time 
that  some  circumpolar  star,  the  nearer  the  pole  the  better,  comes  to  the 
meridian — say  its  upper  passage — turn  the  declination  circle  till  it  reads 
the  star's  polar  distance,  increased  by  the  refraction  due  to  its  altitude,  and 
clamp  the  declination  circle ;  then  by  a  motion  of  the  entire  instrument  in 
right  ascension,  and  the  screws  which  act  upon  the  polar  axis  in  the  me- 
ridian, bring  the  star  to  the  cross  wires,  and  keep  it  there  till  the  instant, 
as  indicated  by  a  time-piece,  of  its  crossing  the  meridian.  This  will  be 
sufficient  for  the  first  approximation. 

Then  observe  some  well-known  star  in  quick  succession  very  near  the 
meridian,  reversing  the  declination  circle.  The  reading  of  the  declination 
circle,  corrected  for  refraction,  will  give,  in  Eq.  (e),  since  <f  =  0, 

p  =  if  -f  4  ir  +  X  .  cos  9, 
p  =  <if,  —  ^  *  -f-  X  .  cos  <p ; 
whence 


The  first  member  being  the  projection  of  the  arc  X  on  the  meridian,  is  the 

19 


290  SPHERICAL   ASTRONOMY. 

arc  by  which  the  pole  is  too  high  or  too  low.  The  axis  being  movod 
through  this  distance  by  estimation,  direct  the  telescope  to  the  polar  dis- 
tance, corrected  for  refraction,  of  a  second  star  soon  to  come  to  the  me- 
ridian ;  when  the  star  is  in  the  field  put  the  clock  movement  in  motion, 
and  as  the  star  culminates,  as  indicated  by  a  time-piece,  bring  the  axial 
wire  to  the  star  by  the  adjusting  screws  of  the  polar  axis  which  are  in  the 
meridian. 

The  polar  distance  of  another  star  when  six  hours  from  the  meridian 
being  observed  in  quick  succession,  in  the  direct  and  reversed  position  of 
the  declination  circle,  Eq.  (e)  gives,  since  in  this  case  tf  =  90°, 

p  =  <ff  +  4  <x  +  X  .  sin  9, 
p  =  iff  —  4  if  -f-  X  sin  9  ; 
whence 


The  first  member  is  the  projection  of  the  arc  X,  on  the  declination  circle  at 
right  angles  to  the  meridian,  and  is,  therefore,  the  deviation  of  the  pole  of 
the  instrument  from  this  latter  plane.  This  error  being  treated  in  a  man- 
ner similar  to  the  preceding,  by  means  of  adjusting  screws  which  act  at 
right  angles  to  the  meridian,  the  polar  axis  is  brought  to  this  latter  plane, 
and  the  instrument  will  be  so  nearly  in  adjustment  as  to  bring  the  errors 
within  the  limitations  .required  to  render  equations  (d)  and  (e)  exact. 

The  approximation  may  be  continued,  if  desirable,  or  the  value  of  each 
error  found  by  recourse  to  celestial  objects  properly  selected,  and  these 
errors  employed  .as  elements  of  reduction. 

To  find  a.  —  Observe  an  equatorial  star  about  the  time  of  its  meridian 
passage,  and  again  after  reversing  the  declination  circle  ;  the  readings  of 
th«  hour  circle  in  Eq  '(d)  give 


whence 


=  (f-±-<f-{-c  cosec  #, 

'  =  tf'  —  12h      4  (f  —  c  cosec  if 


(8  —  (f)  -  (Sf  -  (f'}  -  12h     . 

=  ±  -  -  -  s  -  '-  -  sm  if. 

2 


Denote  by  a  the  right  ascension  of  the  star,  and  by  t  and  t'  the  sidereal 
times  of  observation,  we  have 

s=t  —  a  ;     s'  =  t'  —  a  ; 
in  the  above  equation  give 


APPENDIX    II.  291 

^  (t  —  tf)  —  (tf  —  tf')  —  12h    . 

o  V*'  / 

To  find  1. — Observe  some  star,  near  the  pole,  in  quick  succession  revers- 
ing the  declination  circle ;  the  readings  of  the  hour  circle  in  Eq.  (d)  give 

s  =r  tf  +  ^  tf  -{-  X  sin  (tf  —  9)  cot  if  +  c  .  cosec  tf  +  I  cot  tf, 

*'  =  tf'  —  12h  +  4  (f  +  X  sin  (tf'      12h  —  9)  cot  *  —  c  .  cosec  c  —  I  cot  ir ; 

whence,  since  the  third  terms  of  the  second  members  do  not  differ  sensibly, 

/ .  cot  if  +  c  cosec  if  = — ^- —  —  ; 

eliminating  5  and  s'  by  their  values  t  —  a,  and  t'  —  a,  and  reducing, 


.       \t  —  t'  —  tf  —  tf'  —  12hl .  sin  tf  —  2  c  ,  . 

f-1 2^ir •  •  •  •  w 

To  find  9  anc?  X. — Observe  any  well-known  star,  and  again  after  revers- 
ing the  declination  circle.  The  readings  of  the  circles  in  Equations  (d) 
and  (e)  give 

s  =  tf  +  A  tf  -f  X  .  cot  ve .  sin  (tf  —  9)  +  n, 

^  =  (T7  —  12h  +  A  tf  +  X  .  cot  K  sin  (tf'  —  12h  —  9)  -f-  n', 

^>  =  *  -f-A-^+X.  cos  (tf  —  9), 

in  which 

n  =  c  .  cosec  *  -|-  / .  cot  *, 
n'=  c  .  cosec  ^  +  ^ .  cot  Kr 

Subtracting  the  first  from  the  second,  the  third  from  the  fourth,  transpo- 
sing, and  making,  after  eliminating  sr  and  *  by  their  equals  t'  —  a,  t  —  a, 

2  =  (tr  -   *)  -  (*'  _  <r  _  12h)  -  (n'  -  n), 

we  obtain 

X  ,  cot  <r  [sin  (tfx  —  1 2h  —  9)  —  sin  (tf  —  9)]  =  2  ) 
X  .  [cos  (tfx  -  12h  -  9)  -  cos  (tf  -  9)]  =  H          $ 
but 


'  -  12h  -  0)  -  sin  (<r  -  0)  =  2  sin  |  (ff'-  a  -  12h)  .  cos 


cos  (a'-  12h  -  ^)  -  cos  (a  -  0)  =  2  sin  }  (a'-  <r  -  12h)  .  lin  (a 


°~ 


292  SPHERICAL  ASTRONOMY. 

substituting  these  above,  and  dividing  the  second  equation  by  the  first,  \» 
have,  using  p  for  *, 

/tf'  +  tf—  12h         \       n 
tan  I  --  -  --  9  j  =  —  cot  ;>       ,     .     .     .     (t) 

whence 

o-'  +  er-  12h  n  ,,. 

-  --  ---  tan-' 


. 
and  from  equations  (A)  we  have 


-12h.  su 


To  find  Ati.  —  Observe  a  star  before  its  culmination  in  the  hour  angle 
360°  —  tf,  and  at  an  interval  after  its  culmination  in  the  hour  angle  tf', 
such,  that  360°  —  tf  and  tf'  shall  be  equal,  or  very  nearly  so,  without  re- 
versing the  declination  circle  ;  Eq.  (d)  will  then  give 

24h  —  -  s  =  24h  —  tf  +  A  tf  +  X  cot  if  sin  (360°  —  tf  +  9)  —  n, 
s'  =  0"  +  A  tf  +  X  cot  *  .  sin  (*'  —  (p)  +  n. 

Adding  and  reducing, 

s'  —  s  =  tf'  —  tf  +  2Ao'  +  Xcot'7r'.  [sin  (^  —  9)  —  sin  (tf  +  9)  ]  ; 

writing  sin  (tf  —  (p)  for  sin  (tf7  —  (p),  to  which  it  is  sensibly  equal,  we  have, 
after  developing  the  last  term,  reducing,  and  replacing  *  and  sf  by  their 
equals  t  —  a  and  t'  —  a, 

A  tf  =  —       -  ^—    —  t  4-  X  .  cot  if  .  (sin  9  .  cos  tf)     .     .     (m) 

For  a  star  in  or  near  the  equator,  we  may  take  cot  if  =  0  ;  or  for  a 
star  whose  hour  angle  is  90°,  in  which  case  cos  tf  =  0,  the  above  value  for 
index  error  becomes 


To  find  A  if.  —  Observe  the  same  star  twice  in  quick  succession,  and  in 
reversed  positions  of  the  declination  circle  ;  the  readings  of  the  declinatioh 
circle,  in  Eq.  (e),  give 

p  =  if  +  &<if  +  \  cos  (tf  —  9), 

jt?  =  */--Acr  +  ^  cos  (tf  —  9)  ; 

whence  by  subtraction, 

*'—  if 
4<r~-—      .......    W 


APPENDIX    II.  293 


,       Heliometer. 

1.  —  The  image  formed  by  a  lens  of  a  point  on  .he  surface  of  an  ob- 
ject, is  on  a  line  drawn  through  the  optical  centre  of  the  lens  and  the 
point.     If  the  point  be  stationary  and  the  lens  in  motion,  along  a  line 
perpendicular  to  this  line,  the  image  will  also  be  in  motion,  and  in  the 
same  direction. 

Every  fragment  cut  from  a  lens  by  a  section  parallel  to  its  axis,  forms 
au  image  just  as  large  and  as  perfect  as  does  the  entire  lens,  the  only 
difference  being  in  the  intensity  of  its  illumination,  which  will  be  less  in 
proportion  as  the  surface  of  the  fragment  is  less  than  that  of  the  en- 
tire lens. 

If,  then,  the  lens  be  divided  by  a  plane  through  its  optical  axis,  and  the 
two  halves  moved  in  opposite  directions,  and  perpendicular  to  this  axis, 
an  image  of  an  object  formed  by  the  entire  lens  will  be  duplicated,  and 
the  individuals  of  the  pair  will  be  equally  bright.  Two  half  lenses,  so 
mounted  as  to  be  moved  parallel  to  the  dividing  plane,  called  the  plane 
of  section,  and  at  light  angles  to  the  optical  axis,  by  means  of  micrometer 
screws,  constitute  the  Heliometer.  Such  an  arrangement  forms  the  object- 
glass  of  the  telescope  at  L,  in  Fig.  29.  The  screws  are  furnished  with 
targe  circular  heads,  which  are  carefully  graduated  after  the  manner  ol 
those  of  the  position  micrometer,  and  are  turned  by  the  aid  of  a  rod, 
reaching  to  the  eye-end  of  the  telescope.  The  entire  frame-work,  which 
supports  the  slides  of  the  semi-lenses,  admits  of  a  rotary  motion  about 
the  axis  of  the  telescope's  tube,  and  is  put  in  motion  by  a  second  rod, 
also  passing  to  the  eye-end.  By  this  last  arrangement  the  plane  of  sec- 
tion may  be  made  to  pass  through  any  two  objects,  whose  images  are 
simultaneously  in  the  field  of  view. 

2.  —  The   value  in  arc  of   the  linear  distance  through   which  the 
images  of  the  same  object  are  made  to  separate,  by  turning  the  microm- 
eter screw-head  through  each  unit  of  its  scale,  is  found  by  a  process  in  all 
respects  similar  to  that  explained  in  Appendix  No.  I.,  for  the  position 
micrometer. 

3.  —  Directing  the  telescope  to  the  sun,  duplicating  its  image,  and 
turning  the  micrometer  screws  till  the  images  are  tangent,  the  reading 
multiplied  by  the  angular  value  of  the  head  unit  will  gi  e  the  apparent 
diameter  of  the  sun.  Hence  the  name  of  the  instrument. 


294 


SPHERICAL    ASTROJSOMY. 


4.  —  But  it  is  obvious  that  the  apparent  dimensions  of  any  othe* 
body  may  be  measured  in  the  same  way.  Also  the  angular  distance  sub 
tended  by  the  line,  joining  two  objects,  whose  images  may  be  brought 
into  the  field  of  view  together.  For  this  purpose,  turn  the  whole  field  lens 
till  the  plane  of  section  pass  through  the  objects,  duplicate  the  image  ol 
both,  and  turn  the  micrometer  screws  till  one  of  the  images  of  the  oriu 
be  brought  to  coincide  with  an  image  of  the  other ;  the  reading,  treated 
as  before,  will  give  the  angular  distance  sought. 


The  Sextant. 

1.  —  This  is  employed  in  the  measurement  of  the  angular  distance 
between  two  objects.  It  is  one  of  the  most  generally  useful  instruments 
that  has  yet  been  devised,  furnishing,  as  it  does,  data  for  the  solution  of  a 
variety  of  astronomical  problems  of  the  greatest  practical  utility  both  on 
\and  and  at  sea.  It  is  especially  useful  at  sea,  where  the  unstable  position 
of  the  mariner  excludes  the  use  of  almost  all  other  instruments. 

It  depends  upon  this  catoptrical  principle,  viz.  :  When  a  ray  of  light 
is  reflected  by  two  plane  reflectors  in  a  plane  normal  to  both,  the  ray 
is  deviated  or  bent  from  its  original  direction  through  an  angle  equal  to 
twice  the  angle  made  by  the  reflectors. 

Let  A  C  and  C  B  represent  the  section  of  two  plane  reflectors  perpendic 
ular  to  their  line  of  intersection  C.  RM,  MN,  and  NO  the  course  of  a 
ray  reflected  first  at  the  point  Jf,  and  next  at  the  point  N\  then  will  the 


angle  ROh'  —  'lACB.  For,  draw  the 
normals  M  D,  M  '  D'  to  the  reflector  A  <7, 
nnd  DD'  to  the  reflector  C  B,  and  denote 
by  9  and  9"  the  angles  of  incidence  on  the 
reflector  AC;  by  9'  that  on  the  refiVctor 
C  B,  and  by  i  the  inclination  of  the  reflec- 
tors.  Then,  since  by  the  principle  of  optics 
the  angle  of  incidence  is  equal  to  that  of  re- 
flection, we  have  from  the  triangle  M  DN, 

9  —  9'=i; 
and  trom  the  triangle  M'  D'  N, 

9'- <?"=»•; 

adding  these,  we  have 


Fig.  *>• 


and  because  M  D  and  M'  D'  are  parallel,  the  first  member  is  the  inclina- 
tion of  the  first  incident  to  the  second  reflected  ray. 


APPENDIX   II.  295 

If  then  the  reflector  CB  were  transparent  at  the  point  N,  the  waves  of 
ight  from  an  object  at  R',  would  be  transmitted  through  it  and  coincide  in 
direction  with  those  from  R  reflected  at  M  and  N;  and  to  an  eye  situated 
at  0,  the  objects  R  and  R'  would  apparently  coincide.  Two  reflectors  so 
mounted  as  to  give  the  means  of  reading  their  inclination  to  each  other, 
when  this  coincidence  takes  place,  would  give  the  angular  distance  ROR' 
of  the  objects  by  simple  inspection ;  and,  with  appliances  to  facilitate  the 
operations  of  the  "observer,  constitute  a  reflecting  instrument,  which,  ac- 
cording as  its  arc  of  measurement  is  extended  to  an  entire  circumference  or 
limited  to  an  arc  of  90°,  60°,  or  45°,  is  called  a  reflecting  circle,  quadrant, 
sextant,  or  octant.  The  sextant  is  the  more  common  of  the  instruments 
with  limited  arcs  now  in  use. 

2.  —  The  annexed  figure  represents  a  sextant.  It  consists  of  the  two 
plane-glass  reflectors  C  and  E  seen  edgewise;  a  graduated  arc  A  A,  of 
which  the  plane  is  perpendicular  to  those  of  the  reflectors ;  an  index-arm 
F,  vernier  V,  clamp  and  tangent  screw  0 ;  a  telescope  ED,  of  which  the 
line  of  collirnation  is  parallel  to  the  plane  of  the  arc  of  measurement;  col- 
ored glasses  L  and  K  to  qualify  the  light  received  into  the  telescope,  and 
a  triangular  system  of  frame-work  uniting  strength  with  lightness,  to  sup- 
port all  the  parts  and  render  them  available.  The  handle  of  the  instru- 
ment is  represented  at  H. 

The  arc  of  measurement  is  divided  into  half-degree  spaces,  which  are 
numbered  as  whole  degrees,  and  these  divisions  are  subdivided  to  any  de- 
Fig.  81. 


SPHERICAL    ASTRONOMY. 

feirable  extent  consistent  with  facility  of  reading.  The  reflector  B,  called 
the  index-glass,  is  covered  with  an  amalgam  of  tin  on  the  face  towards  the 
eye-end  of  the  telescope,  and  turns  with  the  index-arm  about  an  axis  in  its 
own  plane,  and  through  the  centre  of  the  arc  <  f  measurement,  being  per- 
pendicular to  the  plane  of  the  latter.  The  reflector  (7,  called  the  horizon- 
glass,  is,  abating  the  limited  range  of  the  adjusting  screws,  securely  fixed 
with  its  plane  also  at  right  angles  to  that  of  the  arc  of  measurement.  Only 
one-half  of  this  glass  is  covered,  and  that  half  lies  nearest  the  frame  of  the 
instrument,  the  covered  face  being  turned  from  the  telescope.  The  line 
separating  the  covered  from  the  uncovered  part  of  this  glass  is  parallel  to 
the  plane  of  the  graduated  arc,  and  at  a  distance  therefrom  about  equal  to 
that  of  the  line  of  collimation,  being  sometimes  a  little  greater  and  some- 
times a  little  less  in  consequence  of  a  change  in  the  position  of  the  tele- 
scope, to  make  the  supply  of  light  it  receives  through  the  uncovered,  equal 
to  that  which  enters  it  after  reflection  from  the  coated  part  of  the  horizon- 
glass.  The  position  of  the  telescope  is  altered  by  means  of  a  screw  and 
milled  nut  connected  with  its  supporting  ring  U.  By  turning  the  nut  the 
telescope  is  thrust  from  or  drawn  towards  the  face  of  the  sextant.  This 
device  is  called  the  up-and-down  piece.  There  are  usually  six  or  seven 
colored  glasses  of  different  shades,  which  are  so  mounted  that  they  can  be 
turned  about  an  axis  c  or  b  parallel  to  the  face  of  the  sextant,  and  be  inter- 
posed or  not  at  pleasure. 

To  facilitate  the  reading,  a  small  microscope  G  is  attached  to  a  swing 
movable  about  an  axis  a,  connected  with  the  index-arm.  Two  telescopes 
and  a  plane  tube,  all  adapted  to  the  ring  C7,  are  packed  with  the  sextant 
One  of  these  telescopes  has  a  greater  magnifying  power  than  the  other,  and 
inverts  the  visible  images  of  objects.  The  telescopes  are  provided  with 
colored  glasses,  which  are  so  mounted  as  to  be  easily  attached  to  the  eye- 
end  to  qualify  the  light  of  the  sun  when  that  body  is  observed. 

Adjustments. 

3. —  The  sextant  requires  three  adjustments, , viz. :  1st.  To  make  the 
index  and  horizon  glasses  perpendicular  to  the  plane  of  the  arc  of  measure- 
ment. 2d.  These  glasses  parallel  to  each  other  when  the  index  is  at  the 
zero  of  the  scale.  3d.  The  optical  axis  of  the  telescope  parallel  to  the 
plane  of  the  arc  of  measurement. 

4.  —  To  accomplish  the  first,  move  the  index  to  the  middle  of  the  arc, 
then  holding  the  instrument  horizontally  with  the  index-glass  towards  the 
eye,  look  obliquely  clown  this  glass  so  as  to  see  the  circular  arc  by  direct 
view  and  by  reflection  at  tb')  same  time.  If  the  arc  appear  broken,  the 


APPENDIX    II.  297 

position  of  the  glass  must  be  altered  till  it  appear  continuous,  by  means  of 
small  screws  that  attach  the  frame  of  the  glass  to  the  instrument. 

The  horizon-glass  is  known  to  be  perpendicular  to  the  plane  of  the  in- 
strument when,  by  a  sweep  of  the  index,  the  reflected  image  of  an  object 
and  the  image  seen  directly,  pass  accurately  over  each  other ;  and  any  er- 
ror is  rectified  by  means  of  an  adjusting  screw,  provided  for  the  purpose,  at 
the  lower  part  of  the  frame  of  the  glass. 

5.  —  The  second  adjustment  is  effected  by  placing  the  index  or  zero 
point  of  the  vernier  to  the  zero  of  the  limb ;  then  directing  the  instrument 
to  some  distant  object  (the  smaller  the  better),  if  it  appear  double,  the  ho- 
rizon-glass must,  after  easing  the  screws  that  attach  it  to  the  instrument,  if 
there  be  no  adjusting  screw  for  the  purpose,  be  turned  around  a  line  in  its 
own  plane  and  perpendicular  to  that  of  the  instrument,  till  the  object  ap- 
pear single;  the  screws  being  tightened,  the  perpendicular  position  of  the 
glass  must  again  be  examined.  The  adjustment  may,  however,  be  rendered 
UD  necessary  by  correcting  an  observation  by  the  index  error.  The  effect 
of  this  error  on  an  angle  measured  by  the  instrument  is  exactly  equal  to 
the  error  itself:  therefore,  in.  modern  instruments,  there  are  seldom  any 
means  applied  for  its  correction,  it  being  considered  preferable  to  determine 
its  amount  previous  to  observing,  or  immediately  after,  and  apply  it  with 
its  proper  sign  to  each  observation.  The  amount  of  the  index  error  may 
be  found  in  the  following  manner :  clamp  the  index  at  about  30  minutes 
to  the  left  of  zero,  and  looking  towards  the  sun,  the  two  images  will  ap- 
pear either  nearly  in  contact  or  overlapping  each  other ;  then  perfect  the 
contact,  by  moving  the  tangent-screw,  and  call  the -minutes  and  seconds 
denoted  by  the  vernier,  the  reading  on  the  arc.  Next  place  the  index 
about  the  same  quantity  to  the  right  of  zero,  or  on  the  arc  of  excess,  and 
make  the  contact  of  the  two  images  perfect  as  before,  and  call  the  minutes 
and  seconds  on  the  arc  of  excess  the  reading  off  the  arc;  half  the  differ- 
ence of  these  numbers  is  the  index  error;  additive  when  the- reading  on  the 
arc  of  excess  is  greater  than  that  on  the  limb,  and  subtractive  when  tbft 
contrary  is  the  case. 

Example. 

i      a 
Reading  on  the  arc       ...     31  56 

"        off  the  arc       ...     31   22 
Difference       .     .     . 
Index  error     .     .     . 


298  SPHERICAL    ASTRONOMY. 

In  this  case  the  reading  on  the  arc  being  greater  than  that  on  the  art 
of  excess,  the  index  error,  =  17  seconds,  must  be  subtracted  from  all  ob- 
servations taken  with  the  instrument,  until  it  be  found,  by  a  similar  pro- 
cess, that  the  index  error  has  altered.  One  observation  on  each  side  of 
zero  is  seldom  considered  enough  to  give  the  index  error  with  sufficient 
exactness  for  particular  purposes  :  it  is  usual  to  take  several  measures  each 
way;  "and  half  the  difference  of  their  means  will  give  a  result  more  to  bb 
depended  on  than  one  deduced  from  a  single  observation  only  on  each 
side  of  zero."  A  proof  of  the  correctness  of  observations  for  index  error  is 
obtained  by  adding  the  above  numbers  together,  and  taking  one-fourth  of 
their  sum,  which  should  be  equal  to  the  sun's  semidiameter,  as  given  in 
the  Nautical  Almanac.  When  the  sun's  altitude  is  low,  not  exceeding  20° 
or  30°,  his  horizontal  instead  of  his  perpendicular  diameter  should  be 
measured  (if  the  observer  intends  to  compare  with  the  Nautical  Almanac, 
otherwise  there  is  no  necessity) ;  because  the  refraction  at  such  an  altitude 
affects  the  lower  border  (or  limb)  more  than  the  upper,  so  as  to  make  his 
perpendicular  diameter  appear  less  than  his  horizontal  one,  which  is  that 
given  in  the  Nautical  Almanac :  in  this  case  the  sextant  must  be  held 
horizontally. 

6.  —  The  third  adjustment  is  made  by  the  aid  of  two  parallel  wires 
placed  in  the  common  focus  of  the  telescope  for  the  purpose  of  directing 
the  observer  to  the  centre  of  the  field  of  view,  in  which  an  observation 
should  always  be  made;  these  wires  are  parallel  to  the  plane  of  the  instru- 
ment, and  divide  the  field  of  view  into  three  nearly  equal  parts.     The  sun 
and  moon  are  made  tangent  to  each  other,  when  their  angular  distance  is 
90°  or  more,  at  one  of  the  wires  ;  the  position  of  the  sextant  is  then  altered 
so  as  to  bring  these  bodies  to  the  second  wire ;  if  the  contact  continue,  the 
line  of  collimation  is  parallel  to  the  plane  of  the  instrument ;  if  not,  the 
position  of  the  telescope  must  be  altered  by  means  of  two  adjusting  screws 
connected  with,  the  up-and-down  piece. 

Artificial  Horizon. 

7.  —  To  measure  directly  the  altitude  of  any  celestial  object  with  the 
sextant,  it  would  be  necessary  that  the  object  and  horizon  should  be  dis- 
tinctly visible ;  but  this  is  not  always  the  case  in  consequence  of  the  irreg- 
ularity of  the  ground  which  conceals  the  hcrizon  from  view.     The  observer 


APPENDIX    II.  299 

Fig.  33. 


is  therefore  obliged  to  have  recourse  to  an  artificial  horizon,  which  consists 
usually  of  the  reflecting  surface  of  some  liquid,  as  mercury  contained  in  a 
small  vessel  A,  which  will  arrange  its  upper  surface  parallel  to  the  natural 
horizon  D AC.  A  ray  of  light  S A,  from  a  star  at  £,  being  incident 
on  the  mercury  at  A,  will  be  reflected  in  the  direction  A  E,  making  the 
angle  SAC=CASr  (AS'  being  EA  produced),  and  the  star  will  ap 
pear  to  an  eye  at  E  as  far  below  the  horizon  as  it  actually  is  above  it 
Now  with  a  sextant  whose  index  and  horizon  glasses  are  represented  at  1 
in.l  H,  the  angle  SES'  may  be  measured ;  but  SES'=SA  S'  —  A  SE, 
and  because  A  E  is  exceedingly  small  as  compared  with  A  S,  the  angle 
A  S  E  may  be  neglected,  and  S  E  S'  will  equal  SA  S',  or  double  the  alti- 
tude of  the  object:  hence  one-half  the  reading  of  the  instrument  will  give 
the  apparent  altitude.  At  sea,  the  observer  has  the  natural  or  sea  horizon 
as  a  point  of  departure,  and  the  altitude  may  be  measured  directly. 

8.  —  Having  now  gone  through  the  principle  and  construction  of  the 
sextant,  it  remains  to  give  some  instructions  as  to  the  manner  of  using  it. 
It  is  evident  that  the  plane  of  the  instrument  must 

Fig  88. 

be  held  in  the  plane  of  the  two  objects,  the  angular 
distance  of  which  is  required.  The  sextant  must  be 
held  in  the  right  hand,  and  as  loosely  as  is  consistent 
with  its  safety,  for  in  grasping  it  too  firmly  the  hand 
is  apt  to  be  rendered  unsteady. 

When  the  altitude  of  an  object,  the  sun  for  instance, 
is  to  be  observed,  the  observer,  having  the  sea-horizon 
before  him,  must  turn  down  one  or  more  of  the  dark 
glasses  or  shades,  according  to  the  brilliancy  of  the  object;  and  directing 
the  telescope  to  that  part  of  the  horizon  immediately  beneath  the  sun,  and 


Fig.  84. 


SPHERICAL    ASTRONOMV 

holding  the  instrument  vertically,  he  must  with  the  left  hand  slide  the 
index  forward,  until  the  image  of  the  sun,  reflected  from  the  index-glass, 
appears  in  contact  with  the  horizon,  seen  through  the  unsilvered  part  of 
the  horizon-glass.  Then  clamp,  and  gently  turn  the  tangent-screw,  to 
make  the  contact  of  the  upper  or  lower  lirnb  of  the  sun  and  the  horizon 
perfect,  when  it  will  appear  a  tangent  to  his  circular  disk.  When  an  arti- 
ficial horizon  is  employed,  the  two  images  of  the  sun  must  be  brought  into 
contact  with  each  other.  To  the  angle  read  off  apply  the  index  error,  and 
then  add  or  subtract  the  sun's  semidiameter,  as  given  in  the  Nautical  Al- 
manac, according  as  the  lower  or  upper  limb  is  observed,  to  obtain  the  ap- 
parent altitude  of  the  sun's  centre. 

The  Principle  of  Repetition. 

1,  —  By  this  principle,  the  invention  of  Borda,  the  error  of  graduation 
Hii  any  instrument  may  be  diminished,  and, 
practically  speaking,  annihilated.  Let  P  Q 
be  two  objects  which  we  may  suppose 
fixed,  for  purposes  of  mere  explanation, 
and  let  0  L  be  a  telescope  movable  on  0, 
the  common  axis  of  two  circles,  A  ML 
and  a  be,  of  which  the  former  A  ML  is 
fixed  in  the  plane  of  the  objects,  and  car- 
ries the  graduations,  and  the  latter  is  free- 
ly movable  on  the  axis.  The  telescope  is 
attached  permanently  to  the  latter  circle, 
and  moves  with  it.  An  arm  OaA  carries 
the  index  or  vernier,  which  reads  off  the 
graduated  limb  of  the  fixed  circle.  This  arm  is  provided  with  two  clamps, 
by  which  it  can  be  temporarily  connected  with  either  circle,  and  detached 
at  pleasure.  Suppose,  now,  the  telescope  directed  to  P.  Clamp  the  index- 
arm  OA  to  the  inner  circle,  and  unclamp  it  from  the  outer,  and  read  orl 
Then  carry  the  telescope  round  to  the  other  object  Q.  In  so  doing,  th«. 
fnner  circle,  and  the  index-arm  which  is  clamped  to  it,  will  also  be  carried 
round,  over  an  arc  A£,  on  the  graduated  limb  of  the  outer,  equal  to  tin 
angle  P  0  Q.  Now  clamp  the  index  to  the  outer  circle,  and  unclamp  tin. 
inner,  and  read  off:  the  difference  of  readings  will  of  course  measure  th«t 
angle  P  0  Q]  but  the  result  will  be  liable  to  two  sources  of  error — that  of 
graduation  and  that  of  observation,  both  of  which  it  is  our  object  to  get 
rid  of.  To  this  end  transfer  the  telescope  back  to  P,  without  unclamping 
the  arm  from  the  outer  circle;  then,  having  made  the  bisection  of  P, 


APPENDIX   II.  301 

clamp  Jie  arm  to  b,  and  unclamp  it  from  J5,  and  again  transfer  1.1m  tele- 
scope to  Q,  by  which  the  arm  will  now  be  carried  with  it  to  GY,  over  a 
second  arc  B  C,  equal  to  the  angle  P  0  Q.  Now  again  read  oft';  then 
will  the  difference  between  this  reading  and  the  original  one  measure  twice 
the  angle  P  0  Q,  affected  with  both  errors  of  observation,  but  only  with  the 
same  error  of  graduation  as  before.  Let  this  process  be  repeated  as  often  as 
we  please  (suppose  ten  times) ;  then  will  the  final  arc  AE  C M  read  oft' on  the 
circle  be  ten  times  the  required  angle,  affected  by  the  joint  errors  of  all  the 
ten  observations,  but  only  by  the  same  constant  error  of  graduation,  which 
depends  on  the  initial  and  final  readings  off  alone. 

The  Reflecting  Circle. 

1.  —  The  use  of  this  instrument  is,  in  general,  the  same  as  that  of 
the  sextant ;  but  when  it  unites,  as  it  often  does,  to  the  catoptrical  prin- 
ciple of  this  latter  instrument,  the  principle  of  repetition,  it  becomes,  in 
the  hands  of  a  skilful  observer,  one  of  the  most  refined  and  elegant  of 
the  portable  implements  in  the  service  of  astronomy. 

This  form  of  the  instrument  is  represented  in  the  annexed  figure. 

The  arc  of  measurement,  which  is  extended  to  the  entire  circum- 
ference, is  divided  into  720.  equal  parts,  and,  for  the  reaton  explained 
in  the  account  of  the  sextant,  these  parts  are  numbered  as  whole  de- 
grees, the  subdivisions  being  continued  to  any  desirable  degree  of  mi- 
nuteness. 

The  circle  is  mounted  upon  two  concentric  axes,  which  may  move  in- 
dependently of  each  other,  and  also  of  the  circle.  Upon  one  end  of  the 


302  SPHERICAL    ASTRONOMY. 

central  axis  is  mounted  a  reflector  E,  similar  to  the  index-glass  of  the 
sextant^  and  upon  the  other  an  arm  A  C,  in  the  position  of  a  diameter  of 
the  circle.  Upon  the  corresponding  ends  of  the  other  axis  are  mounted 
a  system  of  frame-work  and  a  second  arm  B  I).  This  frame-work  sup- 
ports a  second  reflector  F,  similar  to  the  horizon-glass  of  the  sextant,  a 
telescope  If,  colored  glasses  L  and  L\  and  the  handles  /, </,  K  for  hold- 
ing in  different  positions.  The  reflectors  are  perpendicular  to  the  plane 
of  the  circle.  Each  of  the  arms  A  C  and  B  D  has  a  vernier  at  both 
ends,  and  at  one  end  a  vernier,  clamp,  and  tangent-screw,  so  that  the  re- 
flectors may  be  clamped  in  any  position  consistent  with  their  being  per- 
pendicular to  the  plane  of  the  circle,  and  for  each  position  there  will  be 
two  arc  readings,  differing  by  180°. 

At  O  is  seen  the  barrel  for  the  up-and-down  piece,  of  which  the  milled 
head  is  concealed  beneath  the  end  B  of  the  arm  B  D. 

At  M  are  seen  the  microscope  and  its  reflector  for  reading,  mounted 
upon  a  pin  projecting  from  the  vernier  arm. 

The  circle  is  usually  accompanied  by  a  stand,  to  which  it  may  be  at- 
tached, when  great  steadiness  is  required,  by  means  of  screw  holes  in  the 
handles ;  one  of  these  holes  is  seen  in  the  handle  /. 

Adjustments. 

2.  —  The  adjustments  are  the  same  as  those  of  the  sextant,  and 
performed  in  the  same  manner,  with  the  exception  of  the  index  error, 
which,  in  this  instrument,  is  always  eliminated  by  the  manner  of  ob- 
serving. 

Mode  of  Observing. 

3.  —  First  Method. — The  instrument  being  in  adjustment,  clamp  the 
index  A,  and  record  its  reading,  noting  the  degrees,  minutes,  and  seconds 
on  the  vernier  A,  and  the  minutes  and  seconds  on  the  vernier  C.  Un- 
clamp  the  index  Z?,  and  directing  the  telescope  to  one  of  the  two  objects 
whose  angular  distance  is  to  be  measured,  move  the  whole  circle  around 
till  the  two  images  of  this  object  are  brought  nearly  together ;  clamp  the 
index  B,  complete  the  contact  or  coincidence  by  the  tangent-screw,  and 
record  the  reading  as  before,  noting  the  degrees,  minutes,  and  seconds  on 
the  vernier  B,  and  the  minutes  and  seconds  on  the  vernier  D.  The 
glasses  are  now  parallel.  Unclamp  A,  and,  holding  the  circle  in  the 
plane  of  the  objects,  direct  the  telescope  to  the  fainter  of  the  two,  and 
move  the  index  A  till  the  image  of  the  second  object  is  brought  nearly 


APPENDIX   II.  303 

in  contact  with  that  of  the  first ;  clamp,  and  complete  the  contact  by 
the  tangent-screw  :  read  the  verniers  A  and  C  as  before.  The  difference  of 
the  A  readings  will  give  the  angle  as  measured  by  the  sextant,  and  this 
angle  should  always  be  noted  as  a  check.  Next,  unclamp  JB,  and  keeping 
the  telescope  upoo  the  same  object,  move  the  whole  circle  till  the  two 
images  of  this  object  are  again  nearly  in  contact;  clamp,  and  finish  the 
contact  by  the  tangent-screw.  The  glasses  are  again  parallel,  and  the 
index  B  has  passed  over  an  arc  equal  to  the  angular  distance  of  the  two 
objects.  Unclamp  A,  and  move  it  in  the  same  direction  as  before  till 
the  two  objects  again  appear  nearly  in  contact ;  clamp,  and  complete  the 
contact  with  the  tangent-screw ;  the  index  of  A  will  thus  have  passed 
over  an  arc  equal  to  twice  the  angular  distance  of  the  objects.  Now  un- 
clamp B,  and  turn  the  whole  instrument  as  before  till  the  two  images  of 
the  same  object  again  appear ;  clamp,  and  complete  the  contact,  and  the 
index  of  B  will  also  have  passed  over  an  arc  equal  to  twice  the  angular 
distance  of  the  objects.  This  process  being  repeated  as  often  as  may  be 
deemed  desirable,  finally  read  the  verniers  as  before.  Take  a  mean  of 
the  minutes  and  seconds  of  the  first  reading  of  A  and  C,  as  also  of  B 
and  D ;  these  with  the  degrees  of  A  and  B  wHl  give  the  true  readings 
of  the  instrument  at  the  beginning  of  the  operation ;  do  the  same  for  the 
last  reading,  or  that  at  the  close  of  the  repetitions.  Take  the  difference 
between  the  last  and  first  readings  of  the  instrument  for  each  set  of  ver- 
niers ;  add  these  differences  together,  and  divide  the  sum  by  the  numbef 
of  times  that  A  and  B  have  been  moved  after  the  first  contact  of  the  im 
ages  of  the*  same  object :  the  quotient  will  be  the  angle  sought. 

A  comparison  of  this  angle  with  that  given  by  the  difference  of  the 
second  and  (irst  readings  of  A,  will  indicate  the  error,  should  one  have 
been  committed,  either  in  the  readings  or  in  taking  account  of  the  num- 
ber of  repetitions. 

Second  Method. — Clamp  A,  and  record  the  readings  of  A  and  C  as 
before ;  unclamp  B ;  direct  the  telescope  to  the  fainter  of  the  two 
objects,  and  turn  the  circle  till  the  second  object  appear  nearly  in  contact 
with  the  first;  clamp  B\  complete  the  contact  by  the  tangent-screw,  and 
record  the  reading  of  B  and  D.  Now,  invert  the  instrument  by  revolv- 
ing it  through  an  angle  of  180°  about  the  line  of  collimation  of  the  tele- 
scope ;  unclamp  A,  and  move  this  index  till  the  objects  again  appear  nearly 
in  contact ;  clamp,  and  complete  the  contact  by  the  tangent-screw  ;  the 
difference  of  the  second  and  first  readings  of  A  will  be  double  the  angu- 
lar distance  of  the  objects,  the  half  of  which  will  be  the  check.  Bring 
the  instrument  back  to  its  former  position  by  revolving  it  about  the  line 


304:  SPHERICAL    ASTRONOMY. 

of  collimation  ;  unclamp  B,  and  turn  the  circle  till  the  images  agam 
appear ;  clamp,  and  complete  the  contact  by  the  tangent-screw  ;.  the  a™ 
passed  over  by  B  will  also  be  double  that  of  the  objects.  This  process 
being  repeated  as  often  as  the  observer  pleases,  finally  read  the  instrument 
on  both  sets  of  verniers  ;  take  the  first  reading  of  A  and  C  from  the  last  ; 
do  the  same  for  B  and  D ;  add  these  differences  together,  and  divide  the 
sum  by  twice  the  number  of  times  that  A  and  B  have  been  moved  since 
the  first  contact. 

4. —  The  process  of  repeating  is  much  facilitated  by  the  following 
device.  A  brazen  arc  is  attached  to  the  frame-work  of  the  instrument  so 
as  to  be  concentric  with  the  arc  of  measurement,  and  just  below  it,  and 
moves  with  the  telescope  and  horizon-glass.  It  is  out  of  view  in  the  po- 
sition of  the  instrument  represented  in  the  figure.  To  this  arc  are  fitted 
two  small  sliders,  that  adhere  to  it  by  friction,  wherever  placed.  Firmly 
attached  to  the  tangent-screw  end  of  the  arm  A  C  are  two  small  pieces  of 
metal,  called  checks,  lying  in  the  direction  of  radii,  and  just  long  enough 
to  cross  the  brazen  arc,  and  to  slide  over  its  surface,  after  the  manner  that 
the  index  moves  over  the  arc  of  measurement,  so  that  if  one  of  the  sliders 
be  interposed,  the  motion  of  the  index  will  be  arrested. 

5.  —  In  the  first  method  of  observing,  after  the  two  images  of  the 
name  object  are  made  to  coincide,  place  one  of  the  sliders  against  the 
check  on  the  side  from  which  the  index  A  must  be  moved  to  bring  the 
other  object  in  the  field  of  view;  after  the  contact  of  the  two  objects  is 
peifected,  by  moving  the  index  A,  place  the  other  slider  in  contact  with 
the  other  check  on  the  opposite  side.     Now,  the  circle  being  in  the  plane 
ot  the  objects,  a  little  consideration  will  make  it  manifest,  that  to  restore 
the  contact  of  the  images  of  the  same,  and  afterwards  of  the  two  objects, 
it  will  only  be  necessary  to  bring  the  checks  in  contact  with  their  respec- 
tive slides  by  alternately  moving  the  circle  and  index  A.     The  brazen  .arc 
is  sometimes  graduated  and  numbered  in  opposite  directions,  commencing 
from  the  positions  of  the  checks,  corresponding  to  the  parallel  position 
of  the  reflectors ;  this  furnishes  an  additional  check  upon  the  angle  meas- 
ured, and  facilitates  the  management  of  the  sliders.     The  use  of  the  sliders 
in  the  second  method  of  observing  is,  from  what  has  been  said,  too  obvi- 
ous to  need  explanation. 

6.  —  This  Appendix  contains,  it  is  believed,  an  account  of  all  that 
is  essential  in  the  theory,  construction,  and  use  of  the  principal  instru- 
ments employed  in  astronomical  measurements.     To  describe  all  that 
are  in  use,  would  expand  the  work  to  dimensions  inconsistent  with  its 
object,  viz.  :  to  give  to  students  in  the   threshold,  as  it  were,  of  Astro- 


APPENDIX    III. 


305 


nomy,  a  preparation  for  future  progress  in  the  subject.  The  German 
Meridian  Circle  combines  the  Mural  and  Transit,  as  does  also  the 
English  Transit  Circle.  One  of  the  most  useful  instruments  to  which 
the  student  can  give  his  attention,  is  the  Zenith  Telescope,  alluded  to 
on  page  199  of  the  text. 


APPENDIX   III. 


ATMOSPHERIC    REFRACTION. 

When  light  passes  from  one  medium  to  another  it  is  refracted  according 
to  the  law  expressed  by  the  equation,  Optics,  §  15, 

sin  z  =  m  sin  z'     ........     (1) 


in  which  z  is  the  angle  which  the  normal  to  the  inci- 
dent wave  makes  with  the  normal  to  the  deviating 
surface,  and  z'  the  angle  which  the  normal  to  the  de- 
viated wave  makes  with  the  same. 

Denote  the  angle  of  deviation  SAS'  by  r,  then 
will 


which  substituted  •  in  Eq.  (1),  we  have,  after  develop- 
ing, 

sin  z  =  m  (sin  z  cos  r  —  cos  z  sin  r) ; 

and  because  V  is  always  small  when  m  differs  little  from  unity,  which  is 
the  case  in  the  passage  of  light  through  the  different  strata  of  the  atmo- 
sphere, we  may  write 

cos  r  =  1,  and  sin  r  =  r ; 

and  dividing  both  members  of  the  above  equation  by  cos  2,  we  have 

m-1 


r  = 


.  tan  z 


and  regarding  z  as  constant,  r  will  vary  with  ra,  and  hence 


dm 

a  r  =  — -  tan  z 
m* 


(3) 


Now,  if  we  regard  the  atmosphere  as  composed  of  indefinitely  thin  and 
concentric  strata  of  increasing  density  from  the  top  to  the  bottom,  dr  will 
bt  the  deviation  of  the  ray  in  passing  from  one  stratum  to  another  whose 

20 


306 


SPHERICAL   ASTRONOMY. 


indexes  of  retraction  differ  by  dm.  But  in  the  same  kind  of  medium,  this 
difference  is  found  by  careful  experiment  to  be  directly  proportional  to  the 
difference  of  densities  ;  hence 


in  which  X  is  a  constant  and  I)  the  density  of  the  atmosphere  at  the  place 
of  any  one  stratum  whose  index  is  m.     Whence,  Eq.  (3), 


dr  =  X  .  dD  .  tan  z 

Let  0  B  be  the  arc  of  the  earth's  surface  in 
a  vertical  plane  through  a  heavenly  body  S ; 
0' Nf  and  0" N",  two  concentric  strata  of 
atmosphere  in  the  same  plane ;  S  0"  0'  0, 
the  curve  which  is  normal  to  the  front  of  a  lu- 
minous wave  coming  from  the  body  to  the 
observer  at  0.  As  an  object  is  seen  in  the 
direction  of  the  normal  to  the  wave  from  it  as 
the  wave  enters  the  eye,  the  body  will  appear 
to  an  observer  at  0  to  be  at  S',  on  the  line 
tangent  to  the  curve  at  0  ;  and  if  SP  be  the 
prolongation  of  the  straight  portion  of  the  ray 
before  it  enters  the  atmosphere,  the  angle 
S  TSr  will  be  the  total  deviation.  This  angle 
S  T S'  is  called  the  refraction.  Denote  the 
radius  of  the  earth  CO  by  unity;  the  height 
of  the  stratum  N'  0'  above  the  surface  by  x ; 
the  angle  0  C  0'  by  6.  Then  taking  0'  and  0" 
contiguous,  we  have  0'  CO"  =  dQ,  MO"  = 
dx\  and  the  angle  of  incidence  H 0"  S,  on 
the  stratum  of  which  0"  N"  is  the  upper  lim- 
it, being  denoted  by  2,  we  have  MO'  =  dx 
tan  z,  and 

dx  tan  z 


1+x 


(4) 


Fig.  8. 


And  denoting  the  apparent  zenith  distance  Z'  0  S'  by  Z,  we  have 

r=  TPZ'  —  Z  =  6+z~Z\ 
and  by  differentiating 


APPENDIX    Hi.  307 


or  substituting  the  value  of  d&,  in  Eq.  (5), 

dx  .  tan  z 


whence,  Eq.  (4), 

dx  .  tan  z 
•  tang=  +dz; 


dx 


tan  z  1  -f  x ' 

by  integration, 

•      log  sin  2  =  XZ>  -  log  (1  +  *)  +  C; 

and  making  #  =  0,  in  which  case  D  =  D4  and  g  =  Z,  we  have 

log  sin  Z  =XZ>,  -f  (7; 
and  by  subtraction, 

sin  s 

or 

log  -; — ^  =  log  t 
sin  tu 

whence 


But,  Eq.  (4), 

j        -v  j  r»  L  X  sin  2  .  dD 

dr  =  \dD  tan  2  =  —  -- 

Vl  —  sin8  z 
and  substituting  the  value  of  sin  s  above, 


If  the  law  which  connects  the  varying  density  D  with  the  height  x  be 
given,  one  of  these  variables  may  be  eliminated  and  the  integration  per- 
formed. But  in  a  practical  point  of  view  this  is  not  necessaiy  ;  for  X  is 
known  to  be  a  very  small  fraction,  as  is  also  the  greatest  value  of  rr,  the 
latter  not  exceeding  0,01931,  being  the  height  of  the  first  stratum  of  air 
that  has  sensible  action  upon  light,  divided  by  the  radius  of  the  earth,  or 
77  miles  divided  by  4000  miles.  Developing  the  factors  e~*  '  *  '  and 
e  —  2X  (D  —  D)^  negiect,ing  the  second  and  higher  powers  of  X  and  x,  and 
also  the  term  of  which  X  sin8  Z  is  a  factor,  which  may  be  done  without 
sensible  error  when  Z  does  not  exceed  80°,  wi;  find 


308  SPHERICAL    ASTRONOMY. 

X  .  sin  ZdD  X  sin  Z  .  dD 


dr  = 


dr  = 


Vl  +  2x  —  sin2  Z       Vcos2  Z 


_^L^  =  x  tan  Z  .  (1  -  x  sec2  Z)  dD; 

whence 

r  =  X  tan  zf(dD  -  sec8  ZxdD)  • 

and  performing  the  integration,  that  of  the  last  term  by  parts, 
r  =  X  tan  Z  \D  —  sec2  Z  (Dx  — 


but  if  7^  denote  the  height  of  the  mercurial  column  at  any  stratum  of  air 
above  the  observer,  Du  the  density  of  the  mercury,  and  g  the  force  of 
gravity  regarded  as  constant,  then  will 


and 

r  =  X  tan  Z  [D  -  sec2  Z  (Dx  -  Dtih)  +  (7J; 

and  from  the  limit  x  =  0,  where  D  =  D'  and  h  =  h,,  to  the  limit  x  = 
height  of  the  entire  atmosphere,  where  D  =  0,  r  =  0,  and  A  =  0,  we  find 

r  =  X  tan  Z  .  D'  (\  -  h  .  ^  sec2  Z\. 
Taking  the  density  of  Mercury  as  unity,  we  have  the  mean  value  of 


The  mean  value  of  h  is  found  from  the  proportion, 

miles  inches 

4000  :  29.6  :  :  1  :  h: 
which  will  give  for  the  coefficient  of  sec2  Z, 

h.Qi  =  0.0012517. 

Also,  if  D,  be  the  density  of  air  when  the  thermometer  is  50,  and  the  ba- 
rometer 30  inches;  and  we  take  a  =  0.00208,  and  /3  =  0.0001001,  the 
coefficients  of  expansion  for  air  and  mercury  respectively,  then,  Analytical 
Mechanics,  §  245, 

A    1-f  (50-Q./3 
'  '  30  '  1  -f  (t  -  50)  .  a  ' 

in  which  t  denotes  the  actual  temperature  of  the  air  and  mercury  supposed 
the  same,  and  h  the  height  of  the  barometer.  Hence 


APPENDIX    III.  3Q9 

"•*•-  °-0012517  sec'  *  •  •  8 


Had  the  second  power  of  x  been  retained  in  Eq.  (7),  then  would 

r  =  X  D  .  —  .  1H5°~^  ^  -  ^n  Z\  1-0.0012517  sec"  Z+  0.00000139 
30    l-f-(<—  50)  a  \ 


cos*Z 

the  last  term  of  which,  within  the  limits  supposed,  is  insignificant. 
Make 

—  s  •  !tir_"4f  •  ten  z  •  (1  -  °-0012517  sec'  *}  •  •  (9) 

and  we  have 

r  =  \D,u  .........     (10) 

Denote  by  z  and  z'  the  greatest  and  least  observed  zenith  distances  of  a 
oircumpolar  star,  r  and  r'  the  corresponding  refractions,  and  c  the  zenith 
distance  of  the  pole  ;  then  will 


c  = 


In  like  manner,  if  0,  and  z/  be  the  greatest  and  least  zenith  distances  of 
another  circumpolar  star,  r,  and  r/  the  corresponding  refractions, 


2 

Equating  these  values,  replacing  the  refractions  by  the  values  given  in  Eq. 
(10),  we  find 


The  indications  of  the  barometer  and  thermometer  being  substituted  in  Eq. 
(9),  give  u,  u',  ut,  and  «/,  and  therefore  the  value  of  X  J9,.  Numerous 
and  careful  observations  make  XDy  =  5  7  ".82,  which  substituted  in  equa- 
tions (8)  and  (8)',  give  the  refraction  for  every  observed  zenith  distance, 
temperature  of  the  air,  and  height  of  the  barometer. 


310  SPHERICAL    ASTRONOMY. 

APPENDIX    IT. 

SHAPE  AND  DIMENSIONS  OF  THE  EARTH. 

Let  A  MPjA'  represent  a  meridional 
section  of  the  terrestrial  ellipsoid,  M  the 
place  of  the  spectator,  B  the  "north  pole        7j^ 
of  the  earth,  C  its  centre,  Z  the  zenith,      #*' 
HMH'  a  parallel  to  the  rational  horizon 
and  tangent  to  the  meridian  section  at  M^ 
A'  A  the  intersection  of  the  equator  by 
che  meridian  plane. 

Make 

/  =  the  angle  M  GA  =  PKH=  latitude  of  M ; 
A  =  CA,  the  equatorial  radius ; 
B  =  CBj  the  polar  radius. 

Then,  /eferring  the  curve  to  the  centre  and  axis,  its  equation  is 

A?  y*  +  -B2  a?  =  A*B*  (a) 

&  \     / 

the  equation  of  the  tangent  line  HIT, 

A*yy' +  B*xxf  =  A*E* (b) 

and  the  equation  of  the  normal  at  M, 

A*y'(x-x')-J?x'(y-y')  =  0 (c) 

in  which  xf  and  y'  are  the  co-ordinates  of  M. 

Denote  the  angle  MTCby  T,  then  from  Eq.  (b)  we  have 

* 

tan  T=—^-,\ 
A*y 

but  T—  90°  —  I,  whence 

B*x'  tan  /  =  A*yf (d) 

Also,  denoting  the  eccentricity  by  «,  we  have 


Substituting  x' yf  for  xy  in  Eq.  (a),  combining  the  resulting  equation  with 
Eq.  (c?),  and  eliminating  B  by  means  of  Eq.  («),  we  find 


APPENDIX    IV. 
A  cos  / 


311 


" 


VI  -  e2  sin8  / 
,       A  (1  —  ez)  .  sin 

y    ==  TZ==^=Z=T- 

V  1  —  e2  sin2  I 
Differentiating  the  first,  regarding  x  and  I  as  variable,  we  have 

(1  -  e*  sin8 1)% 

but,  designating  by  s  the  linear  dimension  of  any  portion  of  the  arc  of  the 
curve,  we  have  for  the  projection  of  the  element  ds  on  the  axis  of  x, 

ds  .  cos  T  =  ds  .  sin  I; 
and  since  a?  is  a  decreasing  function  of  the  latitude, 

—  dx'  =  ds  .  sin  /; 
which  substituted  in  Eq.  (g)  gives 

ds  =  A  . -      ....         .     (k) 

(1  -  e>  sin2  Z)2 

For  any  other  latitude  I',  we  have 

(1  -  e*  sin8  J')*  ' 
dividing  the  first  by  the  second,  making 

and  solving  with  respect  to  e1,  «re  find 

.       ,    ds-ds' 

*  dssm*l-ds'  sin8  /'   ' 
From  Eq.  (h)  we  have 

and  from  the  well-known  property  of  the  ellipse, 

B  =  A  Vl  —  & (&) 

Making  ds  =  c,  ds'  =  c',  /  =  lm,  V  =  J;m,  we  have  equations  (10)  and 
(11)  of  the  text 


SPHERICAL    ASTRONOMY. 

Denoting  by  R  the  radius  of  curvature  at  any  point  of  the  meridian,  we 
have 


dxd*y        ' 

finding  the  values  of  dv,  dy,  and  d*y  from  Eqs.  (/),  and  substituting 
above,  there  will  result 


(1  -  e*  sin2  I)* 
Then 

Z«R  :  360°  :  :  j3  :  1°; 
whence 


in  which  /3  denotes  the  linear  dimension  of  one  degree  of  latitude. 

Denoting  by  p  the  radius  of  the  earth  in  any  latitude  /,  we  obtain  by 
squaring  and  adding  Eqs.  (/), 


Every  section  of  the  terrestrial  spheroid  through  the  centre  is  an  ellipse  of 
which  the  semi-transverse  and  semi-conjugate  axes  are  respectively  A  and 
p,  /  being  the  latitude  of  the  extremity  of  the  conjugate  axis.  Denoting 
by  ef  the  eccentricity  of  the  elliptical  section,  we  have 


2  _  ^2-p8  _  e8  (1  -  e2)  sin*  I 
'  =        A2  1  -e*  sin2  I 


this  value  of  ef  substituted  in  Eq.  (I)  after  making  therein  /  =  90°,  and 
denoting  by  (3,  the  length  of  a  degree  on  the  section  perpendicular  to  the 
meridian  in  the  latitude  /, 


R  ——     A    V         l~e* sin2 1  (  \ 

P/~360  V:         Y  l-e2(2-e3)sin2Z 

The  value  of  the  radius  of  the  parallel  of  latitude  is  given  by  that  of  a?', 
Eqs.  (/) ;  and  denoting  by  a  the  linear  length  of  a  degree  of  longitude  on 
this  parallel,  we  have 

2  tf  2  if  cos  I 

a  = .  a    = .  A     —         — . ....     (o) 

360  360  v/l-easin8/ 


APPENDIX    V.  313 

Dividing  both  members  of  Eq.  (a)  by  A*  B*,  making  A=  1  and  B  =^ 
that  equation  becomes 

yT  +  **=l  .........     (f) 

Differentiating,  we  find 


but  the  angle  at  Jfin  the  evanescent  triangle  m  Mh  is  equal  to  the  angle 
at  G  =  I'  in  the  triangle  MQD\  and  denoting  in  future  the  central  lati- 
tude M  CD  by  Z,  we  have 

-£«*«>, 

dy 


X 

whence 

tan  I  —  7*  tan  /'     .......     (q) 

Making  A  =  1,  B  =y,  and  eliminating  e2  from  Eq.  (m)  by  the  relation 
£  =  1  —  y8,  we  have 

P  = 


APPENDIX    Y. 

EARTH'S  ORBIT. 

The  sun's  attraction  for  the  earth  varies  inversely  as  the  square  of  the 
distance.  The  earth  describes,  therefore,  an  ellipse  about  the  sun,  having 
the  latter  body  in  one  of  its  foci. 

By  Eq.  (266),  Analytical  Mechanics,  we  have 

da.       2c 


in  which  a  denotes  the  angle  which  the  radius  vector  of  the  earth  makes 
with  any  assumed  axis,  r,  the  radius  vector,  c  the  area  described  by  the  lat* 
ter  in  a  unit  of  time,  and  t  the  time. 


314  SPHERICAL    ASTRONOMY. 

Also,  Eq.  (277),  Analytical  Mechanics, 

a(l-e*)  =  ^       ......    .     (6) 

in  which  a  is  the  semi-transverse  axis  of  the  earth's  orbit,  e  its  eccentricity, 
and  k  the  intensity  of  the  sun's  attraction  on  the  unit  of  mass  of  the  earth 
at  the  unit's  distance. 

The  polar  equation  of  the  ellipse  is 


in  which  V  is  the  true  anomaly,  estimated  from  the  perihelion. 

Eliminating  r  and  c  from  Eq.  (a)  by  means  of  Eqs.  (6)  and  (c),  we  have 


developing  the  factors  of  the  second  members  by  the  binomial  formula, 
and  neglecting  all  the  terms  involving  the  powers  of  e  higher  than  the  sec- 
ond, we  have 

V* 

•^  .  d  t  =  (1  -  |  £)  (1  -  2*  cos  F  +  3e2  cos2  F-  &c.)  da. 
0s 

and  because 

cos8  F  =  £  +  £cos2  F, 


and,  §  201,  Analytical  Mechanics, 


in  which  T  is  the  periodic  time,  and  m  the  mean  daily  motion  of  a  point 
on  the  radius  vector  at  the  unit's  distance  from  the  sun  ;  whence  we  have 


=  da  —  2ecos  VdV+l  e*  cos  2  Vd2  V—  &c.; 
and  by  integration, 

mt+  C=a  —  2esiu  F+f  e2  sin  2F—  &c. 

Making  F  =  0,  and  estimating  a  from  the  line  through  the  vernal  equinox, 
we  have 

mtp+  (7=  a,; 


APPENDIX    V  315 

in  which  a,f  's  the  longitude  of  the  perihelion,  and  tp  the  time  from  peri- 
helion passage.     Whence,  by  subtraction, 

m  (t  —  tp)  =  a  —  ap  —  2  e  sin  V  +  f  e*  sin  2  V—  &c.          .     (e) 
but 

a-ap=F; 
whence 

w  (*  —  tp)=  V—  2esin  F+f  e2  sin  2  7—  &c.       .     .     (g) 

in  which  w  (J  —  ^)  is  the  mean  anomaly,  being  the  mean  angular  dis- 
tance from  perihelion. 

Adding  ap  to  each  member  of  Eq.  (e),  making 

m(t-  tp)  -f  ap  =  aw 
and  writing  a  —  ap  for  F,  we  find 

am  =  a  —  2  e  sin  (a  —  a/>)  +  f  &  sin  2  (a  —  ap)  —  <kc.    .    .    (A) 

in  which  am  is  the  mean,  and  a  the  true  longitude. 

Denote  by  L  the  mean  longitude  at  any  given  epoch,  say  the  beginning 
of  the  year,  and  by  t  the  interval  of  time  since  the  epoch  ;  then  will 


and 

L  +  m  t  =  a  —  2  e  sin  (a  —  af)  +  f  e9  .  sin  2  (a  —  ap)  —  <fec.  .  .  (t) 

Again,  assuming 

^r        cosu—  e  .  ., 

cosF=-  -     .......     (j) 

1  —  e  cos  u 

and  substituting  in  Eq.  (/),  and  replacing  the  first  factor  by  its  value  in 
Eq.  (d),  we  have 

mt+  C  —J  du  (1  —  e  cos  u)  —  u  —  e  sin  u\ 

and  making  t  =  ^  in  which  case  the  body  is  :n  perihelion,  where  F  =  0 
and  therefore  u  —  0,  we  have 

mtp+  (7=0; 
and  by  subtraction,  making  t  —  tp  =  <', 

m  *'=:«  —  e  sin  M       .....    ..(!•) 

From  Eq.  (j)  we  have 

.  tan  -  ; 


310  SPHERICAL    ASTRONOMY. 

i 
from  which  we  have 

V  =  u  +  e  sin  u  +  -  .  sin  2  u (/) 

aud  from  Eq.  (&), 

u  =  m  t'  +  2  e  sin  m  t'  +  -J  e2  sin  2  m  £'  +  &c.  .     .  „   .     (m) 
which  substituted  in  Eq.  (I)  gives 

F=m*'  -|-2e  sin  mtf  +  |  e2  .  sin  2m*'  .     .     .     .     (n) 
whence 

F—  m^  =  2e  sin  m^'  +  |  e2  .  sin  2wi<'  .     .     .     .     (o) 

The  first  member,  which  is  the  difference  between  the  true  and  mean 
anomalies,  is  called  the  equation  of  the  centre.  It  is  expressed  in  terms  of 
the  eccentricity  and  mean  anomaly.  The  auxiliary  angle  u  is  called  the 
eccentric  anomaly. 


APPENDIX    VI 

PLANETS'  ELEMENTS. 
Differentiating  the  equation 

e  cos  v  = 1, 

we  have,  after  dividing  by  d  t, 

dv       L    dr 

e*mv-Tt=l*-dl> 

but,  Analytical  Mechanics,  §  192, 

dv  _  £f 

7i-7' 

which  substituted  above  gives,  after  making 

dr 


is  Eq.  (100)  of  the  text 


APPENDIX   VII. 

APPENDIX    VII 

PLANETS'  ELEMENTS. 
From  Eq.  (277),  Analytical  Mechanics,  we  have 

whence,  making  fx  =  k, 


2c=    f.  vx1  —  «'); 

and  this  in  the  equation 

dv       2c 
Tt=~7' 

Appendix  VI.,  gives 


dt~  * 

and  substituting  the  value  of  r*  from  the  equation 

a  (1  -  f) 


I  +  e  cos  v ' 
we  find 


y'-  (1  +  e  cos  v)f 

To  integrate  this,  assume 

cos  u  —  e 

cos  v  = : 

1  —  e  cos  u 

from  which  find  the  value  of  dv,  eliminate  dv  and  cos  v  above,  and 
have, 

dt  =  — — =  .  (1  —  e  cos  u)  du  ; 
and  by  integration 

t  +  C  =  — —  («  —  e  sin  «). 
But,  Analytical  Mechanics,  §  201, 

a*         T 
Va  ~~  2*' 
in  which  2"  is  the  periodic  time. 


318  SPHERICAL    ASTRONOMY. 

Whence,  making  t  =  0  when  u  =  0,  #e  have  (7=0,  and 

2* 

~  ./  =  «-esm«; 

and  denoting  the  mean  motion  by  n,  we  lave 

2* 

*=T; 

and  finally 

nt  =  M  —  e  sin  «  ;. 

which  is  Eq.  (106)  of  the  text. 

The  quantity  t  is  the  time  from  perihelion,  for  by  malm  g  ti  =  0,  we 
have 

t  =  0 ;     cos  v  =  1,  or  v  =  0°. 


APPENDIX  VIII. 

PLANETS'  ELEMENTS. 
Differentiate  the  equation 

and  divide  by  *2rdt,  we  have 

dr      x    dx      y    dy       z    dz 
dt        r  '  d  t       r  '  dt       r'dt 
and  making 

we  have 


;•?'  +  *  *•  +  ; 


which  is  Eq.  (112)  of  the  text 


APPENDIX    IX.  319 

APPENDIX    IX. 

PLANETS   ELEMENTS. 
Make 

«i,  a*,  a3?  the  observed  right  ascensions; 
^u  &»  A:  the  observed  north  polar  distances ; 

t>t    ^   tai  the  mean  times  of  observations  reduced  to  any  first  meridian,  say 
that  of  Greenwich ; 

and  suppose  the  observed  quantities  corrected  to  the  mean  equinox  and 
mean  position  of  the  equator  at  the  beginning  of  the  year. 

In  the  interval  of  time  required  for  light  to  travel  from  a  roaming  body 
to  the  earth,  the  body  describes  some  definite  portion  of  its  path,  and  at 
any  given  instant  we  see  the  place  it  left  and  not  that  which  it  actually 
occupies.  We  look,  as  it  were,  at  luminous  places  on  the  orbit,  but  always 
behind  the  body's  true  place.  The  position  which  a  body  occupied  at  the 
instani  the  light  started,  and  in  which  it  is  seen  at  a  given  time,  is  called 
its  virtual  place  at  that  time ;  and  that  which  it  actually  occupies  is  called 
its  true  place. 

Conceive  three  sets  of  parallel  rectangular  co-ordinate  axes,  one  set 
through  the  place  of  observation,  another  through  the  centre  of  the  earth, 
and  the  third  through  the  centre  of  the  sun.  Take  the  planes  xy  parallel 
to  the  plane  of  the  equinoctial,  the  axes  of  x  parallel  to  the  line  of  the 
equinoxes  and  positive  towards  the  first  point  of  Aries. 

Denote  by  py  the  distance  of  the  body's  virtual  place  from  the  earth  at 
the  time  tt,  and  by  v  the  time  required  for  light  to  travel  over  the  mean 
radius  of  the  earth's  orbit,  which  we  have  taken  as  unity ;  then  will  v  p;  be 
the  time  required  for  light  to  travel  over  the  distance  p,. 

Denote  by  a?,  y,  z  the  co-ordinates  of  the  virtual,  and  .T,  y,  z  the  co-ordi- 
nates of  the  true  place  of  the  body  at  the  time  /,,  referred  to  the  centre  of 
the  earth ;  then,  regarding  the  motion  of  the  body  as  uniform  during  the 

time  v  p,,  will 

dx 
x  =  x  —  vp,.— 

?  =  ,-.,.£ 
Yl    dt 

dz 


320 


SPHERICAL  ASTRONOMY. 


Denote  the  co-ordinates  of  the  sun,  cleared  of  aberration  at  the  time  /„ 
and  referred  to  the  same  origin,  by  Xt,  Y,,  Z{  ;  and  the  heliocentric  co 
ordinates  of  the  true  place  of  the  body  at  the  same  time  by  xt,  yt,  z,  ;  then 
will 


which  in  Eqs.  (1)  give 


x  =  Xt  +  x,  —  v  P/ 


y  = 


(Z. 


(2) 


in  which 

p,  =  (s,  +  ZJ  sec  /3, (3) 

or,  which  may  be  preferable,  if  the  body  be  near  the  equator, 

py  =  (ary  +  -X))  sec  a,  cosec  /3, (4) 

Denoting  the  co-ordinates  of  the  virtual  place  of  the  body  at  the  time  £,, 
referred  to  the  place  of  observation,  by  »',  y',  z' ;  and  the  co-ordinates 
at  the  same  time  of  the  place  of  observation,  referred  to  the  centre  of  the 
earth,  by/,,  gt,  and  h\  then  will 


~z  =z'  +  A; 
which  substituted  in  Eqs.  (2)  give 


But 


y'  —  x'  tan  a,  =  0 
z   —  x'  tan  0,  =s  0 


(5) 


(6) 


APPENDIX    IX.  321 

in  which  cotan  0,  =  cos  at  .  tan  /3,    .......  (7) 

Also,  if  I  denote  the  geocentric  colatitude  of  the  place  of  observation,  p  the 
corresponding  radius  of  the  earth,  and  Tt  the  sidereal  time  of  observation. 
reduced  to  degrees,  then  will 

ft  =  p  .  sin  I  .  cos  Tt  \ 

^=,p  .sin  /.sin  T  ,  (•    .......  (8) 

h  =  p  .  cos  I  .  ) 

ind 

sun's  horizontal  parallax  at  the  place  of  observation  ' 

~~  number  of  seconds  in  an  arc  equal  in  length  to  radius    '     "  •  •   . 

Multiplying  the  first  of  Eqs.  (5)  by  tan  az  and  subtracting  the  product 
from  the  second,  then  by  tan  0,  and  subtracting  the  product  from  the  third, 
and  reducing  by  the  relations  of  Eqs.  (6),  we  have 


in  like  manner 
y    - 


and 


(10^ 


in  which,  as  in  equations  (3)  and  (4), 


P3  =  (z,  +  Z.)  sec 


or,  if  the  body  be  near  the  equator, 

p2  =  (a*  +  ^2)  sec  «2  .  cosec 
p,  =  (x3  +  JT,)  sec  a,  .  cosec 
Now  make 

<,  =  ^8  —  T,  and  Jj  =  t,  +  rf  ; 
21 


) 
> 


SPHERICAL   ASTRONOMY. 


then,  because  ar,  and  #3  are  functions  of  t,  which  become  2,  when  r  and 
become  zero,  we  have  by  Taylor's  formula, 


*l  -  x*  ~  Tt  ' c  + 

and  the  same  for  y,  and 
,   dx, 


d*xz    «r'8 

_.._ 


_r._    _._ 


and  the  same  for  y3  and  03. 


"(13) 


The  intervals  «r  and  rf  must  be  such  as  to  make  these  expressions  con 
verge  rapidly,  and  it  will  rarely  if  ever  be  necessary  to  retain  the  terms  of 
the  series  involving  powers  of  <r  and  <r'  higher  than  the  fourth. 

Denote  by  p  the  acceleration  due  to  the  sun's  attractive  force  at  the 
mean  distance  of  the  earth,  and  by  rt  the  distance  of  the  body  from  thu? 

sun  at  the  time  *2,  then  will  -^  be  the  acceleration  due  to  the  sun's  attrac- 
ra 

tion  on  the  body,  and  we  shall  have 

d*Xs  fA         #2  _  |Ub2 

7?  ^  ~  rf  '  7.  =  ~"  n1 


y* 


dt* 


(^ 

ra* 


(14) 


Differentiating  and  dividing  by  d  t,  twice,  we  have 
d^Xjj  jtx     dx$       3  jut.    c?7*2 


ra3'  dt 


dra    dxa 


dr* 


and  the  same  for 


,  . 

d?'       dt*  '       dt3  dt*1 


by  simply  writing  y  and  z  successively  for  x. 


APPENDIX    IX.  323 

These  values  being  substituted  in  Eqs.  (13),  and  the  resulting  values  of 

and  also  those  of 

dx\      djji      dzt      d  x*      di/3  dz\ 

y '  ^ a       ar\(\ 

dt  '     dt'    dt'     dt'     dt'          dt' 

obtained  therefrom  in  Eqs.  (10) — observing  to  limit  the  series  to  the  second 
order  of  differentials  in  the  aberration  terms,  or  those  of  which  v  is  a  factor, 
and  to  the  fourth  order  in  the  others — we  have,  after  making 


.     .     .  (15) 


B,  =  (X{  -/,)  tan  4,  -  Z,  +  h 
A9  =  (JT2  — /2)  tan  <%  —  Y,  -f-  g^ 
£s  =  (X2  -/2)  tan  &t  -  Z8  +  h 
A3  =  (JT,  -/,)  tan  a,  -  F3 
^3  =   -T.  --/,)  tan  ^3  _  Z, 


_  t*    X     I  Ur  ^1  f 

*,  =  —  -tana,— 


dX, 


W  — 

*      •  T    . 


dt 

dZ3  .dX3 

=  —r-^  —  tan  d,  — r-^ 


i 
4V 


(16) 


24  r,6 


*-fi? 


dt*        24r2( 


Hr  = 


-'"    drt 


Or,3 


(17) 


324 


SPHLRICAL    ASTRONOMY. 


(          dy2  dx, 

*2  +   __tan«2  — 


(i»; 


and  in  which,  by  substituting  the  values  of  z{  and  z3,  xl  and  #3  in  equations 
(3),  (4),  (11),  and  (12), 


sec 


.     .     .     .  (10) 


or 


p2  =  (x2  +  ^T2)  sec  a2 .  cosec  ft 
«f*t      , 


we  obtain 


(20) 


2 
—  a:2  tan  a,  —  —  <r  +  tan  a:  — 


=  ^,  +  «! 


2,  —  xt  tan  d,  --  -77* 


— ar2tana2 
—  a:  tan  0 


tan  ^,  •—  r 
at 


dv9  dxs    . 

y,  —  a,  tan  a8  +  -^<  —  tan  a,  —  T' 


=  A3  +  a, 


-  x,  tan  03   f-      ST'  -  tan  03      V  = 


(21) 


APPENDIX    IX. 


325 


Regarding  #,,  yf,  zSl  -r^»  -r-^,  and  -j-?  as  unknown  quantities,  we  ob- 


tain bj  elimination, 


(22) 


in  which,  by  making 


R  =Yl  4-  -}  .  sp   **  ~  *'    cos  a»  1 

\          r  /    sin  (a,  —  a3)  cos  as 


j.     ....  (23) 


we  nave 


+  -\    sin  (d»  -  ^i)  c°s  ^3 
r/    sin  (^  —  03)  cos  ^    J 

p        _  COS  a,  COS  a3  r       r'  /      .    <r'  \1 

F    =  TD  -  gn    .     /  —  -  r  •  I  A  --  ±-  A3—  A.2[l  -\  --  II 

(ft  —  S)  sin  (a,  —  a3)     L       r  \          r  /J 

/>  COS  6,  COS  d3  r-    r'  /          r'\-i 

^  =  (Jg-  S)  sin  (*,-».)  '  P-  7  +  ^~  ^l1  +7/J 


cos  a,  cos  a3 


COS    ,  cos 


(24) 


or  by  making 


cos  a,  cos  a, 


-        sn    a- 


cos  d,  cos  ^ 


(R  -  S)  sin  (0,  - 


we  have 


p  =  D  (a,  -  a,)  +  ^  (a,  - 


(25) 


and  making 
we  have 


326 


SPHERICAL    ASTRONOMY. 


G  (68  -  6,)  ^ 


tan 
tan 


~~  =  •—  tan  a,  —  -  (xa  tan  a,  +  A{  +  a,  —  yf 
=         tan  ^  -  i  (a,  tan  dt  +  £,+&,-  »,) 


or  instead  of  the  last  two, 


=  -      tan  a,  +       (or,  tan  a3  +  A,  +  «3  - 


2  =         tan  03  + 


tan  ^  +  ^a  +  6.  - 


'  (26) 


Now  although  the  Eqs.  (26)  express  the  values  of  the  co-ordinates  and 
components  of  the  velocity  of  the  body  at  the  time  of  the  second  observa- 
tion, they  involve  the  geocentric  distances  p,,  p2,  pl?  and  the  radius  vector 
TS,  which  are  unknown,  and  the  solution  of  the  problem  can  only  be  ac- 
complished by  successive  approximations. 

First  Approximation. 

Let  us  first  neglect  the  terms  involving  aberration,  and  those  containing  as 
factors  powers  of  <r  and  T'  higher  than  the  second.  This  will  give,  Eqs.  (17), 

MI  if*  it  «r'^ 


2r23 


and  Eqs.  (18), 


(27) 


APPENDIX   IX. 


327 


and  as  all  terms  involving  powers  of  T  and  <r'  higher  than  the  second  aw 
to  be  neglected  we  obtain  from  Eqs.  (21),  for  the  values  of  the  severa. 
factors  above, 

ys  —  x2  tan  a,  =  A} 

22  —  xa  tan  d ,  —  Bl 

yg  —  #2  tan  ce2  =  A2 

za  —  x2  tan  42  =  B3 

ya  —  #2  tan  a3  =  A3 

which  substituted  in  Eqs.  (27)  give 


=  0; 


3=="" 


and  these  in  the  first  of  Eqs.  (26)  give 


and  making 


we  have 


=  C  + 


N 


(28) 


.    .    .(29) 


.  .(so) 


(31) 


and  this  value  of  xif  and  the  foregoing  values  of  az  and  ba  in  the  second 
and  third  of  Eqs.  (26),  give 

N  tan  a. 


y,  =  C  tan  a,  + 

z,  =  C  tan  6.  + 


JV  tan  d, 


or  making 


C'  =  C'tan  aa  +  ^2  ;         JVT  =  JVtan 


'  =  C'  tan  0, 


'  =  JVtan 


"°  (  '    '         '  (82) 


3BS  SPHERICAL   ASTKONOMY. 


we  have  y,  =  C'  -\ F (33) 


N" 
s,=  <7"  +  £__ (34) 


and  hence 


This  equation  must  be  solved  tentatively.  If  the  body  be  considerably 
more  distant  than  the  earth  from  the  sun,  6'  C",  C"  are  respectively  ap- 
proximations to  the  values  of  x2,  y^  z2  j  and  if  considerably  less  distant, 

N    N'  N" 

then  —  ,  —  -  ,  and  —  3-  are  approximations  to  the  same  quantities.     It  is 

evident  that  a  value  for  r2  must  be  selected  which  will  make  the  greatest 

N  N'  N" 

values  of  C  -\  —  j,    C'  H  --  3,  and  C"  H  --  ^-,  without  regard   to  signs 

,        ,  **2  **2  rS 

r  N 

less  than  r2  and  greater  than  —  L  .     And  if  C  -\  —  ,  ,  for  instance,  be  the 

V3  rs 

greatest,  the  value  of  r2  which  satisfies  the  equation 


will  differ  from  the  true  value  by  a  quantity  less  than  -  I  1  --  =  1  ,  that 

2  \         V37 
is,  less  than  0.21  rs. 

But  the  solution  of  Eq.  (35)  may  be  greatly  facilitated  by  means  of  an 
elegant  geometrical  construction,  due  to  J.  J.  Waterston,  Esq.  Thus  : 

Squaring  the  terms  as  indicated  in  the  second  member,  recalling  the  re- 
lation in  Eq.  (7),  which  will  give 

1  +  tan*  a2  +  tan2  d2  =  tan2  08  sec8  &  ; 
substituting  the  values  of  C",  C",  Nf,  N"  in  Eqs.  (32),  and  making 

r  =  JW  .  tan  02  sec  /32  ; 
K  ==  AZ  tan  at  .  cot  ^2  cos  {32  +  JBa  cos  /3,  ; 
/3  =  K  +  C  tan  02  sec  &  ; 
a'  =  ^2  +  ^22-.JT8; 

equation  (35)  may  be  written 


APPENDIX   IX.  329 


or 
Make 

then  will 


-  =  cosec  &, 


±  -  1  =  cot2  A,     and     -.  =  sins 
a2  r,3 

and  Eq.  (37), 

cot  &  — \-  —.  .  sin1  6 ; 

a        a4 

and  making  cot  d  =  y,  and  sin3  4  ^=  #, 

^    .    r 


Also 

1  +  cot8  6  =  cosec8  d  =   .  .    . 
sin1  6 

or 


3 

whence 


^—7 W 


If  the  curvie  of  which  this  is  the  equation  be  described  graphically,  and  the 
straight  line,  of  which  (38)  is  the  equation,  be  drawn,  the  abscissa  of  their 

point  of  intersection  will  give  the  value  of  sin3  4,  or  —  ;  and  r2  becomes 

rz 
known.     Its  value  may  be  verified  by  substitution  in  Eq.  (35). 

This  value  of  ra  in  Eq.  (31)  gives  xa,  and  this  in  the  second  and  third 
ol  Eqs.  (26)  gives  y.2  and  z8;  also,  r2  in  Eqs.  (29)  gives  a,,  6,,  cr3,  63,  and 
these  in  the  third  of  (25)  will  give  />,  which  with  a,  and  bi  in  fourth,  fifth, 

and  sixth,  or  fourth,  seventh,  and  eighth  of  (26),  give  -~,  -—  ,  and  -^t 
and  the  values  of  p,,  p2,  pi  in  Eqs.  (19)  or  (20). 

Second  Approximation. 
By  differentiating  the  equation 

r28  =  xf  +  yf  +  gf, 
and  dividing  by  r2  dt,  we  have 


ry  _X2     jK*       yf       y^       za        ^ 
dt  ~  r,  '  dt  "*"  r2  '  dt  ^  rt  '  dt 


330 


SPHERICAL    ASTRONOMY. 


The  first  terra  of  this  equation  becoming  thus  known,  the  values  of  £7,  W, 
Ur,  and  TF7,  Eqs.  (17),  may  be  computed  to  include  the  third  powers  of 
r  and  <r',  then  the  corresponding  values  of  al}  6,,  a2,  62,  as,  63,  Eqs.  (18)- 
Then,  denoting  by  A  a,,  A  6,,  Aa2,  A£>2,  Aa3,  A&3  the  difference  between 
the  first  and  second  values  of  the  quantities  written  after  the  symbol  A, 
*ftid  observing  a  like  notation  for  the  other  quantities,  we  have  for  computing 

d  x2      dy%  d  z% 

the  Hrst  corrections  to  #2,  y2,  z2,  —— ,    —:— .  and  — -  from  the  third  and 

at       at  at 

foui  i  of  Eqs.  (25), 


Ap  =  D  (Aa3  —  A«8)  +  E  (Act,  —  Aa2)  ) 
A  q  ==  F  (A  68  -  A  &2)  +  G  (A  6t  —  A  62)  ) 


.     .(41) 


an«     den  from  Eqs.  (26), 

A  xs  =  Ap  —  A  q 

A  y8  =  A  xt  tan  aa  +  A  as 

A  %  =  A  #8  tan  42  +  A  62 


A  — -  =  A  — 7~2  tan  a, (A  x«  tan  a,  -f-  A  a,  —  A  y2) 

at  at  T  ^ 


A         =  A      1  tan  ^  -      (A*2  tan  6,  +  Ab{  -  A*2) 


'    .  .  (42) 


Third  Approximation. 

Differentiating  equation  (40),   dividing  by  d  t,   and   substituting  for 
—r^->    -TT 


-p,    —r^->    -TTi  tn^r  values  in  equations  (14),  we  have 


d? 


df        df 


dt> 


r 


(43) 


r 
with  this  value  for  -j-j  ,  find  new  values  for  Z7",  TF",  IT,  and  IF7  from  Eqs. 

(17);  and  for  a,,  6,,  a2,  &2,  a,3,  J2  from  Eqs.  (18),  by  including  the  terms 
that  were  omitted  before.  Then  with  the  differences  between  these  last 
values  and  the  next  preceding,  form  equations  for  the  final  corrections  by 
writing  A2  for  A  in  equations  (41)  and  (42).  Then  the  final  values  of  the 
required  quantities  become 


APPENDIX    X.  331 


&c.  =  ar, 

&c.  =  y, 
&c.  =  z; 


APPENDIX    X. 

GEOCENTRIC  MOTION. 
By  the  notation  of  the  text,  p.  91, 

a  cos  /  —  cos  Z  =  p  .  cos  X, 
a  sin  /  —  sin  j&  =  p  .  sin  X  ; 
and  by  division, 

a  sin  I  —  sin  L 

tan  X  =  -  -  --  -  ; 
a  cos  /  —  cos  L 

differentiating, 

d  A    _  (a  cos  2  —  cos  L)(a  cos  2  .  dl  —  cos  L  .  dL)  +  (aaml  —  sin  Z)(a  s'ml.dl  —  sin  LdL) 
cos2  A  (a  cos  ^  —  cos  Z)a 

_  [a»—  a  cos  (L  —  1}]  dl+  [1  —  a  cos  (L  —  I)]  dL 
(a  cos  I  —  cos  Z}8 

But  by  Kepler's  3d  law, 

dL  :  dl  ::  a^  :  i; 
whence 

dL  =  a?  .dl; 
which  substituted  above,  and  making 


p  = 


a  cos  ^  —  cos  L  ' 
gives 

d  X:=  P8  .  [a9  +  a?  _  (a  +  J)  cos  (Z  -  /)]  .  rf/; 

and  making 

d\  —  m,     and     dl  =  ny 

we  have  Eq.  (124)  of  the  text. 


332  SPHERICAL    ASTRONOMY 


!        APPENDIX   XI. 

.  ,"  |  ON  ECLIPSLS. 

BY  MR.  W.  8.  B.  WOOLHOU8E,  HEAD  ASSISTANT  ON  THE  NAUTICAL  ALMA  VAC   ESTABLISHMENT. 

Eclipses,  in  all  the  varieties  of  aspect  which  they  present  to  different  places  on 
the  earth,  form  an  entertaining  subject  for  discussion;  and,  without  considering 
the  public  interest  generally  excited  by  their  prediction  and  appearance,  the  use  of 
them,  as  a  test  of  the  degree  of  perfection  of  the  lunar  and  solar  tables,  and  in  the 
determination  and  corroboration  of  geographical  position?,  <fec.,  renders  their  accu- 
rate calculation  an  object  of  some  importance.  The  popularity  of  the  phenomena 
naturally  called  the  attention  of  astronomers,  at  an  early  period,  into  the  field  of 
investigation,  and  several  methods  of  calculation  have  been  adopted  by  different 
authors  at  various  periods. 

For  the  general  circumstances  which  take  place  on  the  earth,  the  plan  of  ortho- 
graphic projection,  though  it  can  only  be  recognized  as  affording  good  approxima- 
tions, seems  to  have  predominated,  and  to  have  been  almost  exclusively  adopted 
in  actual  calculations.  This  method  is  explained  in  the  astronomical  treatises  of 
De  la  Lande  and  Delambre,  and  more  recently  by  Hallaschka,  in  his  JSlementa 
EclipKtwn  (Pragae,  1816),  where  an  example  is  to  be  found  at  length.  Various 
particulars  are -laid  down  in  a  more  accurate  manner  in  Memoires  stir  I'Astronomie 
Pratique.  Par  M.  J.  Monteiro  Da  Rocha,  traduits  du  Portugais  (Paris,  1808). 

The  circumstances  of  an  eclipse  for  a  particular  place  are  usually  calculated  by 
the  "Method  of  the  Nonagesimal,"  which  refers  the  bodies  to  the  ecliptic,  and  an 
example  of  which  may  be  seen  in  the  work  of  Hallaschka  above  mentioned.  This 
part  of  the  subject  has  also  been  discussed  analytically  by  Lagrange,  in  the  Astron. 
Jahrbuch  for  1782;  and  Professor  Bessel  has  since  made  some  important  additions 
to  the  theory,  in  a  paper  inserted  in  the  Astronom/ach*  Naehrichten,  vol.  vii.,  No. 
151,  which  is  to  be  found  translated  in  the  Philosophical  Magazine,  vol.  viii. 

As  the  numerous  calculations  which  may  be  required  for  an  eclipse,  such  as  of 
the  maps,  <fec.,  given  in  the  Nautical  Almanac,  could  not  be  performed  without 
nmny  perplexing  references  to  different  authors,  it  has  been  presumed  that  a  com- 
plete and  systematic  set  of  formulae  would  be  generally  acceptable;  and  such  a 
conviction  has  led  to  the  drawing  up  of  the  fallowing  paper,  which  contains  an 
extensive  classification  of  useful  remarks  and  formulas,  developed  and  arranged 
with  a  careful  view  to  their  practical  application,  and  with  the  endeavor  to  estab 
lish  a  direct  and  uniform  mode  of  conducting  each  species  of  calculation. 


APPENDIX    XI. 


LIMITS  WHICH   DETERMINE  THE   OCCURRENCES   OF   ECLIPSES. 

ELEMENTS. 

The  following  elements,  used  in  the  calculation  of  the  limits,  have  been  derived 
from  the  tables  of  Damoiseau,  Burckhardt,  and  Carlini,  viz.  : 

'      n 

Moon's  horizontal  parallax         .  .      [  greatest         61  32 

)   least  52  50 

Sun's  horizontal  parallax    .  .      \  Sreatest  °     9 

)  least  0     8 

Moon's  semi-diameter  I  Sreatest         16  46 

•  V  least  14  24 

Sun's  semi-diameter.  .         .      I  f  eatest         16  18 

)  least  15  45 

Moon's  hourly  motion  in  longitude    .  I  Sreatest         3^  35 

)   least  27  47 

Sun's  hourly  motion  in  longitude      .  j.  &  r^teat> 

Moon's  hourly  motion  in  latitude  I  freatest          3     4 

least  0     0 


Inclination  of  moon's  orbit  with  ecliptic    .      I  freatest  5°  20     6 

f  least        4     57  22 

LIMITS. 

For  the  occurrence  of  an  eclipse  of  the  moon  : 

1.  The  greatest  possible  distance  of  the  centres  of  the  moon  and  earth's  shadow 
at  the  time  of  contact,  is  63'  29". 

2.  At  the  time  of  true  ecliptic  conjunction  of  the  moon  and  earth's  shadow,  or 
at  the  time  of  opposition  or  full  moon,  the  greatest  possible  latitude  of  the  moon  is 
63'  45". 

3.  At  the  time  of  opposition,  or  full  moon,  the  greatest  possible  distance  of  the 
centre  of  the  moon  or  of  the  earth's  shadow  from  the  ascending  or  descending  node 
of  the  moon's  orbit  is  12°  24'. 

For  the  occurrence  of  an  eclipse  of  the  sun  : 

1.  The  greatest  possible  distance  of  the  centres  of  the  sun  and  moon,  at  the  time 
of  contact,  is  1°  34'  28''. 

2.  At  the  time  of  true  conjunction  of  the  sun  and  moon,  the  greatest  possible 
latitude  of  the  moon  is  1°  34'  52". 

3.  At  the  time  of  true  conjunction  of  the  sun   and  moon,  or  the  time  of  new 
moon,  the  greatest  possible  distance  of  the  centre  of  the  sun  or  moon  from  one  of 
the  nodes  of  the  moon's  orbit  is  18°  36'. 

The  third  of  these  limits  applies  to  the  true  place  of  the  node,  which  may  differ 
considerably  from  the  mean  place. 

The  most  convenient  and  certain  limits,  however,  will  be  those  of  the  moon'f 
latitude  (/?),  and  will  be  as  follows  : 

1.  At  the  time  of  full  moon  an  eclipse  of  the  moon  will  be 
certain         )      ,        .,   (    <  51'  57'1 
impossible  P    JDM   >6345 
and  doubtful  between  these  limits. 


334:  SPHERICAL    ASTRONOMY. 

For  the  doubtful  cases,  an  eclipse  will  result  when 


in  which  P,  s  denote  the  equatorial  horizontal  parallax  and  semi-diameter  of  the 
moon,  and  T,  o  those  of  the  sun. 

2.  At  the  time  of  new  moon  an  eclipse  of  the  sun  will  be 

certain         {  when/JJ    <1°23'15' 
impossible   >  I    >  1     34    52 

and  doubtful  between  these  limits. 

For  the  doubtful  cases,  an  eclipse  will  happen  when 
&  <(,?_-  *)+,+**  +  25' 

PARALLAX. 

If  a  straight  line  be  drawn  from  the  centre  of  the  earth  to  any  assumed  place,  it 
will  be  the  radius  of  the  earth  for  that  place,  and  this  radius  we  shall  designate  by 
the  letter  p.  This  radius  p,  produced  upward  towards  the  heavens,  will  determine 
what  we  shall  call  the  central  zenith,  being  that  point  which  spherically  deter- 
mines our  true  position  in  relation  to  the  centre  of  the  earth.  The  apparent  ze- 
nith, however,  is  naturally  determined  by  a  line  which  is  vertical  to  the  observer, 
and  therefore  a  normal  to  the  spheroidal  surface  of  the  earth.  The  small  angular 
deviation  of  this  normal  from  the  radius  of  the  earth,  or  the  angular  distance  be- 
tween the  central  and  apparent  zeniths,  is  what  astronomers  call  "  the  angle  ol 
the  vertical  ;"  and,  the  earth  being  an  oblate  spheroid,  it  is  evident  that  the  cen- 
tral zenith  will  be  nearer  to  the  equator  than  the  apparent,  and  also  that  the  hor- 
izontal parallax  will  always  be  less  than  that  at  the  equator,  in  consequence  of  the 
diminution  of  the  earth's  radius  in  proceeding  towards  the  poles.  The  effect  of 
parallax  on  the  position  of  a  body  above  the  horizon  is  to  augment  its  zenith  dis- 
tance, and  for  this  we  have  the  well-known  relation, 

"  sin  par.  in  zun.  dist.  =  sin  hor.  par.  X  sin  app.  zen.  dist." 

This  relation  will  hold  strictly  for  the  spheroidal  figure  of  the  earth,  provided  we 
adopt  the  central  zenith,  and  that  horizontal  parallax  which  appertains  to  the  ra- 
dius p  of  the  place  of  observation. 

Consider  the  equatorial  semi-diameter  of  the  earth  as  unity,  and  let  y  denote 
the  polar  semi-diameter,  which,  adopting  the  mean  between  La  Lande  and  Delam 

304 

bre,  will  be  -  .     Let  also  /  be  the  latitude  of  the  central  zenith,  or  what  is  usu- 
305 

ally  called  the  "  geocentric  latitude,"  and  I  that  of  the  apparent  zenith,  which  may 
be  termed  the  spheroidal  or  geographical  latitude.  Then  the  co-ordinates  of  this 
place,  referred,  in  the  plane  of  its  meridian,  to  the  polar  axis,  will  be 

x  =  p  sin  I,  y  =  p  cos  I. 

By  the  generating  ellipse 


and  therefore  for  the  angle  T,  which  the  normal  makes  with  y  or  the  tangent  with 
«,  we  have 

dy        1     x        tan  I 
tan  I1  =  —  -—  =s  —  .  -  =  —  —  , 
dx      Y'    y          ya 

.-.  tan  /  =  y  tan  I'      .     .  .     . 


APPENDIX    XI.  335 

Again,  the  values  of  x  and  y,  substituted  in  the  above  equation  of  the  ellipse, 
give 


and  hence 

"  ' 


To  these  may  be  added  the  following,  which  are  sometimes  useful,  and  directly 
deducible  from  the  equations  (1),  (2), 

y*  tan  /'  (1  -  *')  sin  /' 

x  =  a  sin  /  =  =  =     ,  _  =•      ....     (3) 

Vl-fy2  tauu/'        VI-**  sin*  I' 

1  cos  I' 

y  =  p  cos  I  =  -  =  ....     (4) 

_  V  1  -f  y'  tan2  *'       VI  -  e*  sin'  f 

where  <?=  v/1  —  ya  is  the  eccentricity  of  the  meridian. 
Also        * 


The  equations  (1),  (2)  are  convenient,  and  the  latter  may  be  simply  resolved  by 
logarithms,  thus : 


p  =  COS  \j/ 

From  (1)  may  also  be  deduced 

tan  x  =  y  tan  I'  =s =  v  tan  I  tan  I' 

tan  (/'  — -  I)  ^  — ; sin  2  \ 

Here  we  may  remark,  that  in  reducing  the  geographical  latitude  to  the  geocen- 
tric with  the  argument  Z',  the  auxiliary  arc  X,  being  between  the  values  of  I  and  /'. 
will  be  a  very  small  quantity  in  defect  of  the  argument ;  and  that,  on  the  contrary 
in  reducing  the  geocentric  to  the  geographical  latitude,  the  arc  x  ^ill  exceed  the 
argument  by  nearly  the  same  quantity.  Therefore,  if  we  assume  x  as  an  argu- 
ment for  the  difference  I'  —  I,  a  table  formed  from  the  equation 

/I  -  Y*\ 

tan  (I'  —  /)=:( —  I  sin  2  x> 


/'  -  /  =  (2y1~nyi,,)  «n  2  x,  in  seconds, 

will  be  equally  adapted  to  both  reductions,  giving  nearly  the  mean  between  them ; 
and  a  table  so  constructed,  with  the  argument  x»  signifying  either  latitude,  will 
answer  every  necessary  degree  of  accuracy,  since  the  reduction  itself  is  so  small 
fn  numbers  we  have 

-~^-'  = — , and  its  logarithm  =  7.51641  .'.log  ( ^—^—  )  =r  2.83084, 

2y        2X304X306  6\2ytanl"/ 

and  hence 

I'  -  /=  [2.83084]  sin  2X. 


336  SPHERICAL   ASTRONOMY. 

Thus  the  following  table  has  been  derived : 


Difference  between  the  Geographical  and  Geocentric 

Latitudes. 

Argument:  %,  either  Latitude. 

X 

V—  I 

X 

I'—  I 

i 

X 

I  -I 

0     0 

,  „ 

o    o 

i  ,i 

o    o 

, 

o  90 

c   c 

i5  75 

5  39 

3o  60 

9  47 

I   89 

c  24 

16  74 

5  59 

3i  59 

9  58 

2  88 

o  47 

17  73 

6  19 

32  58 

10   9 

3  87 

i  ii 

18  72 

6  38 

33  57 

10  19 

4  86 

i  34 

19  71 

6  57 

34  56 

10  28 

5  85 

i  58 

20  70 

7  i5 

35  55 

10  37 

6  84 

2   21 

21   69 

7  33 

36  54 

10  44 

7  83 

2  44 

22   68 

7  5i 

37  53 

10  5  1 

8  82 

3   7 

23   67 

8   7 

38  52 

10   57 

9  81 

3  29 

24  66 

8  23 

39  5i 

n   3 

10  80 

3  5a 

25  65 

8  39 

4o  5o 

ii   7 

ii  79 

4  i4 

26  64 

8  54 

4  1  49 

ii   ii 

12  78 

4  36 

27  63 

9   8 

42  48 

ii  14 

i3  77 

4  57 

28  62 

9  22 

43  47 

ii  16 

i4  76 

5  18 

29  61 

9  34 

44  46 

ii  17 

i5  75 

5  39 

3o  60 

9  47 

45  45 

ii  17 

The  difference  is  to  be  subtracted  from  the  geographical,  or  added  to  the  geo- 
centric latitude,  whether  it  be  north  or  south. 

It  is  evident  from  what  has  been  said,  page  334,  that  if  Z  denote  the  true  dis- 
tance of  the  moon  from  the  central  zenith  as  it  would  appear  at  the  centre  of  the 
earth,  and  Z'  the  apparent  distance  from  the  same  zenith,  as  seen  from  the  place 
on  the  surface,  where  the  radius  of  the  earth  is  p  ;  and  furthermore,  P  the  equato- 
rial horizontal  parallax,  and  z  =  Z'  —  Z,  the  parallax  in  altitude,  we  shall  have 


sin  z=/»  sin  P  sin  Z' 

Substituting  Z  -f-  z  in  the  place  of  Z',  and  dividing  by  cos  z,  we  find 

p  sin  P  sin  Z 


tan  z  = 


1  —  p  sin  P  cos  Z 
which  are  the  usual  formulae  for  the  parallax  in  altitude. 
For  the  radius  p  of  tne  earth  we  have  log 


(8) 


(9) 


—  va 

JL.  =  8.909435,  and      by  (6) 


tan  $  =  [8.909435]  sin  /, 
p  =  cos  $. 

The  values  of  p  so  computed  are  given  in  the  annexed  table. 


APPENDIX    XI. 


337 


Log.  Radius  of  the  Earth. 
Argument:  'Geocentric  Latitude. 

I 

%  P 

I 

log  f 

J 

log  f 

0 

o 

o 

O 

o  .00000 

3o 

9.99964 

60 

9.99893 

I 

o  .00000 

3i 

9.99962 

61 

9.99891 

a 

o  •  ooooo 

32 

9-99960 

62 

9.99889 

3 

o  .  ooooo 

33 

9.99958 

63 

9.99887 

4 

9.99999 

34 

9-99955 

64 

9.99885 

5 

9.99999 

35 

9-99953 

65 

9.99883 

6 

9.99998 

36 

9-99951 

66 

9.99881 

7 

9.99998 

37 

9-99948 

67 

9-99879 

8 

9.99997 

38 

9-99946 

68 

9.99877 

9 

9.99997 

39 

9.99943 

69 

9.99876 

10 

9.99996 

4o 

9.9994r 

70 

9.99874 

ii 

9.99995 

4i 

9-99938 

71 

9.99872 

12 

9.99994 

42 

9-999^6 

72 

9.99871 

i3 

9.99993 

43 

9.999^4 

73 

9.99870 

i4 

9.99992 

44 

9.999^1 

74 

9.99868 

i5 

9.99990 

45 

9.99929 

75 

9.99867 

16 

9.99989 

46 

9.99926 

76 

9.99866 

'7 

9.99988 

47 

9.99924 

77 

9.99866 

18 

9.99986 

48 

9.99921 

7« 

9.99864 

'9 

9-99985 

49 

9.99919 

79 

9.99863 

20 

9.99983 

5o 

9.99916 

80 

9.99862 

21 

9.99982 

5i 

9.99914 

8r           9.99861 

22 

9.99980 

52 

9.99911 

82 

9.99860 

23 

9.99978 

53 

9.99909 

83 

9.99859 

24 

9.99976 

54 

9.99907 

84 

9.99859 

25 

9.99974 

55 

9.99904 

85  ' 

9-99858 

26 

9.99973 

56 

9.99902 

86 

9.99858 

27 

9.9997! 

57 

9.99900 

87 

9.99858 

28 

9-99968 

58 

9.99897 

88 

9.99858 

o9 

9-99966 

59 

9.99895 

89 

9.99857 

3o 

9.99964 

60 

9.99893 

9° 

9.99857 

PHENOMENA  WHICH  TAKE    PLACE    ON   THE   EARTH   GENERALl  Y. 

The  place  on  the  surface  of  the  earth  where  the  limbs  of  the  sun  and  mocn  first 
appear  in  contact  will  be  where  the  penumbra  first  touches  the  earth,  and,  conse- 
quently, at  this  place  the  apparent  contact  will  be  in  the  horizon,  the  disk  of  the 
moon  being  wholly  above  the  horizon,  and  that  of  the  sun  below  it.  The  point  of 
contact  will  l>e  in  the  same  vertical  with  the  two  centres;  and,  therefore,  the  real 
as  well  as  the  apparent  places  will  be  in  the  same  vertical  circle ;  and  the  lower 
limb  of  the  moon,  being  in  the  horizon,  will  be  depressed  by  the  whole  amount  of 
the  horizontal  parallax  which  belongs  at  that  time  to  the  latitude  of  the  place. 
Similarly,  the  place  which  first  has  a  cr ntral  eclipse  will  be  where  the  straight  line 
through  the  centres  of  the  sun  and  moon  comes  first  in  contact  with  the  earth,  and 
at  this  place  the  centres  of  both  objects  will  be  in  the  horizon,  that  of  the  moon 
experiencing  the  whole  effect  of  the  horizontal  parallax. 

22 


338  SPHERICAL    ASTRONOMY. 

The  same  circumstances  will  have  place  where  the  phenomena  finally  quit  the 
earth. 

Since  the  apparent  places  of  the  sun  and  moon  are  so  contiguous,  and  the  par- 
allax of  the  sun  so  small,  it  is  evident  that  the  relative  positions  will  be  the  same 
if  we  give  to  the  moon  the  effect  of  the  difference  of  the  parallaxes  P  —  it,  and 
retain  the  sun  in  his  true  position.  This  difference  P  —  it  is  therefore  the  relative 
parallax,  or  that  which  influences  the  relative  position  of  the  bodies.  If  p  be  the 
radius  of  the  earth  for  the  place  on  its  surface,  the  parallax  which  ought  to  be 
used  ie  p  (P  —  it).  But  in  the  following  investigations,  where  a  place  is  generally 
the  object  of  determination,  we  cannot  previously  so  reduce  this  relative  parallax 
P  —  it.  In  order  therefore  to  secure  the  chance  of  least  deviation  from  the  truth 
in  this  respect,  we  shall  in  these  cases  reduce  the  parallax  in  the  first  instance  to 
a  mean  latitude  of  45°,  so  that  it  will  be  [9.99929]  (P  —  v).  We  shall  conse- 
quently, to  simplify  the  analytical  expressions,  hereafter  denote  this  quantity  by 
the  letter  P'  only  ;  except  in  one  or  two  instances,  where  the  latitude  of  the  place 
is  known,  and  where  it  is  always  distinctly  specified  to  represent  the  parallax 
properly  reduced  to  that  latitude,  or  p  (P  —  *•). 

I    PLACES  WHERE  THE  DIFFERENT  PHASES  ARE  FIRST  AND  LAST  SEEN  ON 
THE  EARTH. 

Let  the  whole  be  referred  to  the  sur-     .  Fig.  5. 

face  of  a  sphere  concentric  with  the 
earth  ;  and  let  0  R  be  the  relative  orbit 
of  the  moon,  which  is  generated  by  the 
differences  of  the  motions  in  right  as- 
cension and  declination,  or  by  the  rela- 
tive  motion  of  the  moon  ;  N  ihe  north 
pole  ;  S  the  sun  ;  Sn  perpendicular  to 

the  relative  orbit,  the  nearest  approach         v ^ 

which    we  denote  by  n\  G  the    point  ^T 

where  the  rnoon  comes  in  conjunction  in 

right  ascension,  and  OS  the  difference  of  declination  at  that  time,  which  we 
denote  by  contraction,  diff.  dec.  Let  also  MM'  be  the  positions  of  the  moon,  when 
a  distance  of  the  centres  equal  to  A'  first  appears  on,  and  finally  quits  the  eaith  ; 
M S=M'  S=  A,  the  corresponding  true  distance  as  seen  from  the  centre  of  tho 
earth  ;  ZZ'  the  zeniths  of  these  places  on  the  earth,  which  must  be  respectively 
in  the  continuations  of  SM,  8  M1,  in  order  that  the  full  effect  of  parallax  may  be 
communicated  in  causing  the  bodies  to  approach. 

As  the  apparent  zenith  distance  of  the  points  which  experience  the  greatest 
effect  must  be  90°,  we  may  evidently  assume  Z$  =  90°:  for  contact  of  either 
limb  of  the  moon  with  the  contiguous  limb  of  the  sun,  we  have  accurately 
Z/S  =  (90°  —  TT)  -f-  a  ;  for  contact  of  either  limb  of  the  moon  with  the  remote  limb 
of  the  sun  ZS=(90°  -*)  —  »;  and  for  contact  of  the  centres  Z£=90°  —  *. 
By  making  Z  $  =  90°,  the  phase  will  begin  with  sunrise  and  end  with  sunset ;  and 
it  is  evident  that  no  sensible  augmentation  can  affect  the  semi-diameter  of  the 
moon  so  near  the  horizon.  The  true  distance  S  M  of  the  centres  being  A,  and  f* 
the  relative  horizontal  parallax,  the  apparent  distance  A'  will  be  P'  ~  A  ;  and 
by  estimating  positive  distances  from  S  towards  M,  in  order  to  have  the  first  oc- 
currence of  the  phase,  it  will  be  A  —  P' ; 

.-.    A=P'-f  A'. 


APPENDIX   XT. 

Here  we  may  notice  three  limiting  aspects,  — 

(1)  When  simple  or  exterior  contact  of  limbs  first  takes  place, 

A'  =  *  -f  ff,  and  A  =  Pr  +  s  -f-  a- 

(2)  When  interior  contact  of  limbs  first  takes  place    A'  =  s  ~  ff;  when  *  >  r.  a 

total  contact  first  commences  with  A  '  =  »  —  a  ;  when  s  <  o,  an  annular  con 
tact  first  commences  with  A  '  =  a  —  s.     Therefore, 
If  s  >  <7,  a  total  eclipse  first  begins  on  the  earth,  when 


If  s  <  <r,  an  annular  eclipse  first  begins  on  the  earth,  when 

A  =P'-s+cr. 
(3)  When  contact  of  centres  first  takes  place  on  the  earth, 

A'  =  0  and  A  =P'. 

For  the  time  of  true  conjunction  in  right  ascension,  assume 
D,  the  true  declination  of  the  moon  ; 
a,  the  true  difference  of  right  ascension,  or  D  's  right  ascension  minus  0's 

right  ascension,  in  space  ; 
Di,  the  relative  motion  in  declination,  or  the  motion  of  the  moon  in  declina- 

tion, minus  that  of  the  sun,  at  that  time  ; 
fll,  the  relative  motion  in  right  ascension  at  the  same  time  ; 
t,  the  inclination  of  the  relative  orbit   0  R  with  a  parallel  of  declination 

through  the  point  O,  or  the  angle  CSn; 

•>,  the  angle  under  the  distance  and  the  line  of  nearest  approach,  or  the  angle 
MS  n.    This  angle  is  always  measured  on  the  northern  side  of  the  dis- 
tance, so  that  when  0  R  falls  below  S,  or  when  diff.  dec.  C  8  is  negative, 
it  will  exceed  90°. 
Then  the  relations  of  the  figure  will  give  these  equations  : 

tant  =  -  1—=;        n  =  (diff.  dec.)  cos  <  ......     (1) 

ii  cos  D 

Di 

Hourly  motion  in  the  orbit  =  -  —  , 
sin  i 

arc  n  G  =  n  tan  t. 

For  the  time  of  describing  the  arc  n  (7,  or  the  interval  between  the  middle  of 
the  general  eclipse  and  the  time  of  conjunction,  it  must  be  divided  by  the  hourly 
motion  in  the  orbit.  Therefore,  t  denoting  this  interval, 


tan  c. 
Assume 


(n  sin  t 
~BT 


.  =  3600"  X  =  [8.66630] 


.....    <2> 
t  in  seconds  =  c  tan  i  J 

The  sign  of    will  be  determined  by  combining  the  signs  of  diff  dec.  and  D\  ; 
and  then 

time  of  middle  =  time  of  <5  —  t  ........    (3) 

Also 

cos«  =  —   ...........     (4) 

A 

Mn  =  n  tan  w. 


310  SPHERICAL   ASTRONOMY. 

Let  T  denote  the  semi-duration  of  the  phase,  or  the  time  of  describing  Mn,  and 

T  in  seconds  =  c  tan 
i   u~. 
Time  < 


Again,  let,  at  the  beginning,  the  /  NSZ  =  a,  and  for  the  ending,  the 
/  N  S  Z' =•  6;  and,  these  angles  being  estimated  from  N8  towards  the  east,  we 
Bhall  have 

«  =  (-,)_«,  &  =  (-,)  +  „ (6) 

and,  the  sun  being  supposed  in  the  horizon,  Z  S  =  90°,  Z'  8=  90°, 


cos  NZ  =cosNSZ  sinNS,        tanZNS=-  ^^—r, 

cos  NS 

cos  NZ'  =  cosN8Z'  sin  NS,       tanZ'NS  =  -  i&nNSZ'. 

cos  N8 

tan  a 

em  I  =  cos  a  cos  6  ;       tan  h  =  — 

sin  <5 


tan  6 
sin  /'  =  cos  b  cos  5  ;       tan  h'  = : — - 

sin  S 

the  latitude  and  hour  angle  /,  h,  relating  to  the  first  place,  and  /',  h',  to  the  last. 
These  hour  angles  are  measured  from  the  sun  towards  the  east,  so  that  the  longi- 
tudes of  the  places  will  be  determined  by  subtracting  respectively  from  them  the 
apparent  Greenwich  times  of  beginning  and  ending  reduced  into  degrees  and  min- 
utes, observing  that  positive  differences  will  indicate  east  longitudes  and  negative 
differences  west  longitudes. 

In  the  preceding  formulas  we  must  use, 

f  Partial    ~]  f  P'  -f  s  +  <r, 

For  beginning  and       Total  P'  +  s  —  a, 

«nding  of  a         1   Annular   f  EcllPse'  A  =  \    P' -  s  +  *, 

{.  Central    J  I  P'. 


II.  RISING  AND  SETTING  LIMITS. 

The  places  ZZr,  thus  found,  are  the  two  extreme  points  of  a  series  of  places 
"where,  at  the  intermediate  times,  the  same  phase  will  appear  in  the  horizon ;  and 
for  the  phase  of  external  contact  of  limbs,  the  curves  which  these  places  assume 
form  one  of  the  principal  geographical  limits  of  the  general  eclipse.  In  the  an- 
nexed diagram,  let  M  be  the  place  of  the 

moon    at   a   time    between  the  beginning,  w 

and  ending  of  the  partial  eclipse.  Make 
Sw=A',  Jfm  =  P',  and  m  Z=  90°  ; 
then  at  the  place  Z  the  moon  will  appear 
at  m,  and  have  simple  external  contact 
with  the  sun  in  the  horizon.  The  two  tri- 
angles SmM,  Sm'  M,  will  give  two  such 
places  at  each  instant,  which,  on  consider- 
ing the  passage  of  the  penumbra  over  the 
terrestrial  disk,  evidently  ought  to  be  the 


APPENDIX    XI.  341 

case.  Since  Mm  =  P'  and  8 m  =  A',  the  possibility  of  forming  the  triangles 
8  m  M,  S  m'  M,  will  depend  on  two  conditions  for  the  value  of  S  M,  viz., 
8 M  <  Mm  +  Sm,  SM>  Mm  -  Sm,  or  A  <  Pf  -f  A'  and  >  P'  —  A',  that 
is,  A  must  be  between  the  values  P'  —  A'  and  P'-f-  A'  :  this  leads  to  two  spe- 
cies of  curves. 


1.    When  the  nearest  approach  is  greater  than  P'  —  A '. 

Here  the  formation  of  the  triangles  Sm  M,  8m' M,  will  always  be  possible  du- 
ring the  appearance  of  the  phase  on  the  earth.  At  the  first  appearance  and  final 
departure  of  the  phase,  8  M  =  Mm  -f-  Sm,  the  triangle  Sm  M  will  be  simply  the 
line  S  M,  and  only  one  place  Z  will  result.  By  taking  positions  of  M  on  both  sides 
of  the  middle  point  n,  it  will  also  appear  that  the  relative  positions  of  the  places 
Z  Zf  become  inverted,  and  that  the  curves  described  by  them  must  intersect  each 
other  at  some  intermediate  place.  Hence  it  appears  that  the  curve  of  risings  and 
settings  commences  with  a  single  point,  which  immediately  after  divides  itself  into 
two  points  moving  in  opposite  directions  on  the  earth,  and  which  describe  two 
curves  intersecting  each  other,  and  finally  meeting  again  in  a  single  point,  the  whole 
forming  one  continued  curve,  returning  into  itself,  and  assuming  the  figure  of  an 
8  much  distorted.  At  the  place  where  they  intersect,  the  phase  will  begin  at  sun 
rise  and  end  at  sunset,  or  it  will  begin  at  sunset  and  end  at  sunrise. 

2.    When  th,e  nearest  approach  is  less  than  P'  —  A '. 

In  this  case  the  triangles  SmM,  Sm'  M,  will  resolve  into  the  line  $  Jtf"  when 
A  =  P'  +  A '  and  also  when  A  =  P'  —  A ',  each  of  which  positions  will  give  only 
one  place  Z,  Thus  it  appears  that  the  points  Z  will  form  two  distinct,  oval,  and 
isolated  curves,  the  former  curve  being  generated  between  the  decreasing  values 
A  =P/-f-  A'  and  A  =  P'  —  A'-,  and  the  latter  between  the  increasing  values 
A  =  P' —  A 'and  A  =  P'  -f-  A'.  The  leading  point  of  the  first  oval  and  the 
terminating  point  of  the  second  oval  are  the  places  where  the  phase  begins  and 
ends  on  the  earth.  The  terminating  point  of  the  first  oval  and  the  leading  point 
of  the  second  oval  are  simply  determined  by  using  A  =  P'  —  A',  and  computing 
the  same  as  for  the  beginning  and  ending  of  a  phase  on  the  earth. 

Let  us  now  turn  our  attention  to  the  determination  of  the  two  places  Z  Z',  at 
any  time,  or  for  any  position  of  M.  Join  Z  S  and  draw  Md  perpendicular  to 
NS. 

We  shall,  throughout  our  investigation,  usually  denote  Sdby  (x),  d  M  by  (y\ 
and  the  /  dS  Mby  S,  this  angle  being  estimated  from  S  JV  towards  the  east. 

To  determine  these  quantities,  let  the  declination  of  the  point  d=(D),  which 
will  a  little  exceed  that  of  M,  and  which  is  distinguished  from  it  by  being  placed 
within  a  parenthesis;  then,  supposing  N  M  to  be  joined,  the  right-angled  spherical 

triangle  Nd  M  will  give  tan  (D)  = .     As  a  is  always  small,  the  difference  of 

the  declinations  (D)  —  D-=  tan-1 D  maybe  arranged  in  a  small  table 

M  annexed 


342 


SPHERICAL    ASTRONOMY. 


Difference  between  (D]  and  D,  or  a  corr. 

Arguments  :  D  and  a. 

a 

D 

10 

20 

3o 

4o 

5o 

60 

70 

80 

00 

IOO 

o 

„ 

„ 

,, 

„ 

„ 

„ 

„ 

„ 

„ 

II 

O 

O 

o 

o 

3 

o 

o 

0 

o 

O 

o 

I 

o 

0 

0 

O 

o 

i 

I 

i 

I 

2 

2 

o 

o. 

o 

0 

i 

i 

I 

2 

2 

3 

3 

o 

0 

0 

I 

i 

2 

2 

3 

4 

5 

4 

o 

o 

I 

2 

2 

3 

4 

5 

6 

5 

0 

0 

j 

2 

3 

4 

5 

6 

8 

6 

o 

o 

I 

2 

3 

4 

6 

7 

9 

7 

o 

0 

2 

3 

4 

5 

7 

9 

ii 

8 

o 

o 

2 

3 

4 

6 

8 

10 

12 

9 

o 

2 

3 

5 

7 

9 

ii 

i3 

10 

o 

2 

4 

5 

7 

10 

12 

i5 

ii 

c 

3 

4 

6 

8 

10 

i3 

16 

12 

0 

2 

3 

4 

6 

9 

ii 

i4 

18  i 

i3 

o 

2 

3 

5 

7 

9 

12 

i5 

19  i 

i4 

o 

2 

3 

5 

7 

10 

i3 

17 

20   ! 

t5 

o 

2 

3 

5 

8 

ii 

14 

18 

22 

16 

o 

2 

4 

6 

8 

ii 

i5 

I9 

23 

17 

0 

2 

4 

6 

9 

12 

16 

20 

24 

18 

o 

2 

4 

6 

9 

1  3 

16 

21 

26 

19 

o 

2 

4 

7 

10 

i3 

17 

22 

27 

20 

0 

'  3 

4 

7 

TO 

i4 

18 

23 

28 

21 

o 

3 

5 

7 

TI 

i4 

IQ 

24 

29 

22 

o 

3 

5 

8 

II 

i5 

IQ 

25 

3o 

23 

o 

3 

5 

8 

II 

i5 

20 

25 

3i 

24 

0 

3 

5 

8 

12 

16 

21 

26 

32 

25 

o 

3 

5 

8 

12 

16 

21 

27 

33 

26 

o 

3 

6 

9 

12 

17 

22 

28 

34 

27 

0 

3 

6 

9 

13 

17 

23 

29 

35 

28 

o 

3 

6 

9 

13 

18 

23 

29 

36 

29 

o 

3 

6 

9 

i3 

18 

24 

3o 

37 

The  number  of  seconds  given  by  this  table,  which  we  have  denoted  by  the 
term  a  corr.,  is  to  be  applied  so  as  to  increase  D,  whether  it  be  north  or  south. 

The  value  of  (Z>)  being  found  by  so  correcting  D  with  this  table,  we  shall  evi- 
dently have 


--8 


__ 

cos  8' 


(A) 


the  quadrant  in  which  S  is  to  be  taken  being  determined  by  (x)  and  (y)  as  co- 
ordinates. 


APPENDIX   XI.  343 

We  shall  afterwards  have  frequent  occasion  to  use  these  quantities. 

If  t  denote  the  time  from  the  middle  of  the  general  eclipse,  they  may  be  deter- 
mined more  easily,  though  less  accurately,  by  means  of  the  following  formulae, 
which  may  readily  be  inferred  from  what  has  preceded. 


t 
tan  w  =  -,  A  = 


(B) 


e  cos  w ' 

8--=  (-4*0, 
(x)  =  A  cos  S,  (y)  =  A  sin  8,  . 

the  upper  sign  being  for  the  time  t  before  the  middle,  and  the  under  sign  for  the 
same  time  after  the  middle. 

Denote  the  /  mMS  by  m.  In  the  triangle  mMS,  which  may,  on  account  of 
its  smallness,  be  considered  as  a  plane  one,  we  also  have  Jf/w=P',  Sm=  A'. 
and  S  Jt/  =  A.  Assume 


_P'—  A' 


and  then 


P'  .   A 

As  ZS,Zm  maybe  considered  as  quadrantal  arcs,  they  will  be  parallel  at 
the  extremities  S,  m;  and  thus  the  /  ZSM~  /.  mMS~m.  Therefore  the 
Z  N 8 Z—  S  ±  m ;  and  the  sun  being  supposed  in  the  horizon,  the  spherical  tri- 
angle NSZ  will  have  ZS=9V°,  and  hence  the  places  Z,  Z',  will  depend  on  the 
following  formulae,  in  which  Z  is  called  the  place  advancing,  and  Z'  the  place  fol 
lowing. 

Place  following, 

sin  I  =  cos  (S  —  m)  cos  J.  tan  h  = *      .  "T      , 

sin  3  ,  .  . 

Place  advancing,  |  ' 

tan  (8  4-  m) 

em  I  =  cos  (S  +  m)  cos  <5,  tan  A  = V— i , 

sin  o 

In  these  expressions  the  symbol  S  represents  the  declination  of  the  sun  at  the 
time  for  which  we  calculate ;  but  for  common  purposes  the  value  of  S  at  the  time 
of  conjunction  may  be  used  in  all  cases. 

III.  NORTHERN  AND  SOUTHERN  LIMITS  FOR  ANY  PHASE. 

The  determination  of  the  extreme  latitudinal  limits  of  a  phase,  or  of  the  terres 
trial  lines  whereon  that  phase  will  appear  as  the  middle  of  the  local  eclipse,  is  the 
most  complex  and  unmanageable  of  all  operations  which  relate  to  a  general  eclipse. 
For  any  given  phase,  at  different  places  on  the  earth,  the  moon  must  be  so  reduced 
by  parallax  as  to  touc'i  a  given  concentric  circle  on  the  solar  disk  ;  and  if  we  con 
sider  this  circle,  by  way  of  illustration,  to  represent,  instead  of  the  sun,  the  disk 
of  the  luminous  body,  the  places  on  the  earth  which  severally  see  the  given  phase 
must  be  situated  in  the  surface  of  the  penumbral  or  umbral  cone,  according  as  the 
interfering  limb  of  the  moon  only  approaches  or  projects  over  the  centre  of  the 
sun;  that  is,  the  places  must  all  be  found  in  the  intersection  of  this  cone  with  the 
surface  of  the  earth.  This  intersection  will  assume  a  complete  or  partial  oval 


ojffiERICAL    ASTRONOMY. 

lorm,  according  as  the  cone  falls  wholly  or  partially  on  the  earth's  illuminated 
disk.  When  it  falls  only  partially  on  the  earth,  the  extreme  points  will  evidently 
see  the  sun  in  the  horizon,  and  be  therefore  two  points  belonging  to  the  horizon 
limits ;  -but  in  the  other  case  the  phase  cannot  at  that  instant  be  seen  in  the  hori- 
zon. It  is  evident  then,  that  these  two  cases  have  been  already  characterized  in 
the  discussion  of  the  rising  and  setting  limits.  Let  us  now  suppose  the  bodies  to 
assume  consecutive  positions,  answering  to  very  small  intervals  of  time,  the  earth 
also  turning  round  its  axis,  and  we  shall  have  a  series  of  these  ovals.  It  is  obvious 
that  the  extreme  geographical  limits  of  the  phase  will  be  represented  by  curves 
which  envelope  all  these  ovals; — that  at  each  instant  the  place  of  limit,  by  reason 
of  the  compound  of  the  motions,  will  be  proceeding  relatively  in  the  direction  <>J 
the  tangent  to  the  oval ; — that  there  will  be  two  of  these  limits  when  the  oval 
becomes  entire  during  the  eclipse,  but  only  one  when  it  is  always  partial.  This  is 
the  most  popular  and  natural  idea  that  can  be  formed  of  the  nature  of  these  limits ; 
and  we  may  here  remark,  as  an  inference  from  what  has  been  said,  that  if  the 
rising  and  setting  limits  of  any  phase  do  not  extend  throughout  the  general  partial 
eclipse,  there  will  be  both  a  northern  and  southern  limit  to  that  phase ;  but  that, 
on  the  contrary,  when  the  rising  and  setting  limits  continue  throughout  the  eclipse, 
there  will  be  only  one  of  these  limits  to  the  phase,  viz. :  a  southern  limit  when  the 
difference  of  declination  at  conjunction  is  positive,  and- a  northern  one  when  that 
difference  is  negative. 

As  before,  let  the  system  be  referred  to  a 
sphere  concentric  with  the  earth,  and  let  M  be 
the  place  of  the  moon ;  Z,  Z',  the  zeniths  of  the 
places  which  are  respectively  in  the  northern 
and  southern  limits ;  and  m,  m',  the  corres- 
ponding apparent  places  of  the  moon.  Draw 
the  meridians  N  m',  NS,  N  m,  N  Z,  NZ',; 
aiso  m  r,  m'  r',  and  M  h  d  h'  perpendicular  to 
ATS;  and  assume  Sd=(x),dM=(y),m  h  =  x, 

/_  N  rn  Z—M,  /  m  N  S  =  a',  declination  of 
m  =  D',  and  the  latitude  of  Z=l.  Then  the  /  mNZ—h  —  a\  m  M—Z1  Bin  Z, 
x  =  mMco&  M—P'  sin  Z  cos  M  and  y  =  m  M  sin  M  =  P'  sin  Z  sin  M ;  these  by 
spherics  resolve  thus : 

x  =  P'  sin  Z  cos  M 

=  P1  [sin  /  cos  D'  ->-  cos  /  sin  Df  cos  (h  —a')  ] 
y  =  P'  sin  Z  sin  M 

=  P'  cos  /  sin  (A  —  a') 
From  these  we  deduce 

««=«—(*) 

—  P'  sin  Z  cos  M —  (x) 

=  P'  [sin  /  cos  D'  —  cos  /  sin  £'  cos  (h  —  a')]  —  (x)  , 


=  (y)  —  P'  cos  I  sin  (h  —  a') 

L«t  us  now  keep  our  attention  to  the  same  place  Z  on  the  earth,  and  suppose 
the  system  to  be  in  motion  as  in  nature.     The  hour  angle  h  will  increase  at  th« 


APPENDIX    XI.  34-5 

rate  of  15°  per  hour,  and  the  latitude  /  will  by  hypothesis  remain  unchanged;  *t 
that  the  following  equations  will  ensue  : 

—  -  =  —  P1  sin  1"  -j-  [sin  I  sin  D'  -f  cos  I  cos  D'  cos  (h  —  a')  ] 
d  t  at 


+  P'  sin  l'Yl5°  —  ^\  cos  /  sin  D'  sin  (h  —  «')  —  ^ 
V  a  t  /  at 

=  —  P'  sin  1"  d-^-  cos  Z+  P  sin  I"(l5°  —  ~)  sin  D'  sinZsin  Jf—  ^ 
at  \  or  /  dt 

±!L  =  ^}  _  P'  8ia  1"  (150  -  %)  cos  /  cos  (A  -  a') 
a£         a  £  \  at/ 

=  1^)  __  p'  8in  1"  (i  6°  —  ^-\  (cos  Z  cos  £'  —  sin  Z  sin  D'  cos  M  ). 
at  \  at/ 


Now,  in  order  that  m  may  be  the  apparent  place  of  the  moon  at  the  middle  of 
the  eclipse,  and  consequently  her  nearest  apparent  contiguity  with  the  sun,  we 
must  have 

—  —  =  0  ;  or  since  u*-{-v*—  A  /2,  u  —  —  (-  v  —  =  0,    which   is   the   condition    of 
a  t  at         at 

limit 

.    Before  we  substitute  the  preceding  values  of  -  —  ,  —  ,  it  may  be  observed,  to 

at    dt 

avoid  complexity,  that  the  quantities  P1  sin  1''  —  —  ,  P1  sin  1"  -  —  may  be  neg- 

d  t  d  t 

l°cted  as  being  very  small  compared  with  P'.  15°  sin  1",  ~—  '  and  -^  ;  also  that 

&  may  be  substituted  for  D',  which  will  equally  serve  the  purpose  of  both  northern 
and  southern  limits.     With  these  modifications  we  have 

~  =  P'.  15°  sin  1"  sin  S  sin  Z  sin  M  —  ^ 


=  _  p,   15o  8in  !/,  (C08  ^  cos  5  —  sin  Z  sin  i  cos  M  ) 

at         dt 

and,  for  the  condition  of  limit, 

u  [pr.  15°  sin  1"  sin  J  sin  Z  sin  M  —  - 

+  v  l^~  —  P'.  15°  sin  1"  (cos  /  cos  i  —  sin  Z  sin  S  cos  JIf  )1  =  0. 

Instead  of  P1  sin  Z  cos  M  put  (a;)  -f-  w.  and  for  .P'  sin  Z  sin  Jf  put  (y)  —  v,  and 
it  becomes 


Fl50  sin  1"  (y)  sin  t  —          J  +  v  Fl50  sin  1"  (*)  sin  t  +  ^ 

—  P'  v  16°  sin  1"  cos  Z  cos  6  =  0  ; 


. 

*.  cos  Z 


346  SPHERICAL   ASTRONOMY. 

But,  if  ai  denote  the  true  relative  motion  in  right  ascension,  and  Di  the  true 
relative  motion  in  declination,  and  D  the  declination  of  the  moon,  at  the  time  ol 
true  conjunction, 


.-.  cosZ  = 

V  COS  C 

Make  now  the  following  assumptions  : 

ii  cos  D 

(O) 


(3) 


P1  cos  5 

in  which  (A),  (£)  may  be  used  as  constant  quantities  throughout  the  eclipse,  and 
we  get 

cos  Z  =  —  (  —  u  sin  v  -j-  v  cos  v). 

v 

The  angle  r  S  m  is  equal  to  the  inclination  of  the  apparent  relative  orbit  with  the 
parallel  of  declination;  denote  it  by  t',  and  then  u  =  A'  cos  t',  v  =  A'  sin  i',  and 


(4) 


which  is  a  concise  form  of  the  condition  to  be  fulfilled  by  Z  and  t',  in  order  that 
the  place  Z  may  be  situated  in  the  limit  of  a  phase. 

Since  the  /  MS  d  —  S,  and  the  Z  M  S  m  =  180°  —  (8  +  *'),  /.  MSm'  — 
S~{-  i,  we  have  for  the  triangle  MSm 

Mm*  =  A2+  A'2±  2  A  A'  cos  (8  +  «')• 
Divide  this  by  P'8  and  we  get 


for  the  geometrical  relation  between  S  and  t',  the  upper  sign  applying  to  the 
northern,  and  the  under  sign  to  the  southern  limit.  Add  this  to  the  square  of  the 
preceding  equation  (4),  and  there  results 


for  the  determination  of  the  angle  t'. 

The  solution  of  this  equation  is  by  no  means  very  practicable  ;  but  as  a  small 
error  in  the  value  of  Z  will  not  sensibly  affect  the  angle  i',  we  may  have  recourse 
to  the  following  indirect  process,  in  which  we  first  consider  the  angle  i'  to  be 
equal  to  i,  which  in  most  instances  is  very  nearly  so.  The  letter  M  designates  the 
the  angle  Mm  h. 


APPENDIX    XI. 


347 


M  =  A     SOS  I 

v  =  £  '  sin  t 


tttJf— £ 


(D)  =  D  +  (a  —  a')  corr. 

(a  —  a')  COS  (D)  Z  =  (Z>)  -—  D' 

= *_ __.  y 

P'  sin  Jf 


P'  cos  Jf 
the  upper  signs  being  for  the  northern,  and  the  under  signs  for  the  southern  limit 

Or,  if  t  be  the  time  from  the  middle  of  the  general  eclipse,  and  w'  the  angle 
under  Mm  and  the  line  of  nearest  approach,  we  shall  have 

Mm  sin  u'  =  n  tan  u>  =  n  — ,  and  Mm  cos  «'  =  n  ±   A', 

which,  observing  that  Mm  =  P'  sin  Z,  give  the  following  equations,  wherein  E 
and  F  are  constant  for  all  the  computations. 


n  ±  A 


e(n  ±  A') 


tan  a/  = 


northern 
southern 

F 


I  limi 


limit. 


cos 


*>=(  — 


•  (8) 


before  )  ,, 
after     f th 

The  sign  of  the  constants  JE,  F,  are  the  same  as  that  of  n  ±  A'  ;    and  when  this  is 
negative,  the  angle  w'  will  be  in  the  second  quadrant. 

The  value  of  Z  determined  in  this  manner  will  be  sufficiently  approximate  for 
the  purposes  of  a  general  map;  and  where  greater  minuteness  is  wanted,  it  will 
serve  very  well  to  get  the  angle  i  from  the  equation  (4).  For  this  we  have 

COB  Z 


cot  *'  =  cot  v  — 


A  sin  v 


which  may  be  resolved  thus : 


tan 


2*  cosv  tan'=^^ (9> 

After  t'  is  so  found,  which  is  only  wanted  roughly,  the  accuracy  of  the  calculation 
may  be  tested  by  the  equation  (4) ;  and  then  we  may  proceed  to  a  correct  compu- 
tation of  M  Z,  by  the  equations  (7),  only  using  t'  instead  of  i.  We  shall  thus 
have  in  the  sphorical  triangle  Zm  N,  ZM=Z,  Nm  =  90°  —  D',  and  the  angle 
Zm  N=M;  and  I  V  spherics  the  following  formulae: 

tan  9  =  tan  Z  cos  M 


tan  (h  —  a') 


=  cos(0  +  ^)ta 
check 


sin  0 


tan  j  =  tan 

sin  Z  cos  M 


CO8   ft  — 


(10 


cos  (e  -f  D'}       cos  (h  —  a')  cos  I 


For  a  map  the  equations  (8)  and  (10)  will  alone  be  amply  sufficient.  In  fact, 
where  a  very  accurate  calculation  is  wanted,  the  most  satisfactory  method  will 
consist  in  first  computing  the  places  roughly ;  then  to  reduce  the  horizontal  paral- 
lax to  the  latitude  by  means  of  the  radius  p,  from  the  table  at  page  337,  and  with 


34:8  SPHERICAL    ASTRONOMY. 

the  use  of  the  value  of  Z,  to  find  the  augmented  semUdiameter  of  the  moon  by 
means  of  the  table  at  page  360,  and  thence  the  proper  value  of  A  ',  and  then  to 
follow  the  equations  (3),  (9),  (4),  (7),  (10). 

The  first  and  last  points  of  these  limits  will  have  Z=  90°.  For  these  places  we 
have  therefore  by  (5) 

P"=  A2  +  A'2±'2  A   A'cos(S+i'). 

If  we  assume  i  =  t,  we  shall  obviously  have  -8  +  *'  =  $+»=&>,  and 
A  cos  (8  +  i)  =n,  w  being  the  angle  under  the  distance  A  and  the  nearest  ap- 
proach n,  as  before  used. 

.'.  P'a=  A3+  A'2±  2  A'n» 


Consequently 

A2sin2«  =  A2  —  w2  =  P'3  —  (n±  A')2, 

which  divided  by  A  2  cos2  o»  =  w2,  gives 


tan  «  =  -  V  P'2  —  (n  ±  A  ')*. 

Therefore  by  taking  the  constant  c  used  in  the  computation  of  the  beginning 
and  ending  of  a  phase  on  the  earth,  we  shall  have 

semi-duration  =  c  tan  <,,  =  —  v  P/z  —  (w  ±  A  ')a, 

M» 

which  may  be  arranged  for  calculation  as  follows : 

n  ±  A'  P'    .      ,    -} 

cos  w  =  — — — ,  semi-duration  =  c  —  sin  w ,    j 

Time  of  |  ^e^ture  [  =  iime  °f  middle  |  +  }  8emi-duration> 

The  places  of  entrance  and  departure  of  the  limits,  by  continuing  the  assump- 
tion «'  =  «,  may  be  hence  calculated  as  for  the  beginning  and  ending  of  a  phase 
only  using  8  ^  u  instead  of  S,  thus : 

&  ^  u  =  D't 

For  place  of  entrance, 

tan  a        I  /io 

sin  I  =  cos  a  cos  D',          tan  h  =. : — =-7,     r  •  •  \»* 

sm  Df 

For  place  of  departure, 

sin  /=  cos  b  cos  D',          tan  A  =  — 


sin.0" 

Having  assumed  t'  =  »,  the  times  and  places  so  computed  will  only  be  approxi- 
mate, though  sufficiently  near  for  general  purposes.  For  an  accurate  calculation, 
we  must  first  determine  the  true  value  of  ir.  Since  Z=90°,  the  equations  (9) 
give  t'  =  v,  which  is  also  shown  by  (4).  We  may,  therefore,  with  the  quantities 
taken  out  for  the  respective  times  of  entrance  and  departure,  proceed  with  the 
equations  (C),  (3),  use  v  instead  of  t  in  (7),  and  then  the  final  results  will  be  deter- 
mined by  (10).  It  ought,  however,  to  be  observed,  that  it  will  be  advisable  to 
take  the  time  of  entrance  in  excess  to  the  next  higher  integral  minute,  and  to  re- 
ject fractions  of  a  minute  in  the  time  of  departure ;  since  by  fixing  on  a  time  a 
trifle  without  the  actual  limits,  the  value  of  sin  Z  would  come  out  greater  than 


APPENDIX    X  349 

unity,  and  the  calculation  rendered  useless  in  consequence.  The  places  so  compu- 
ted will  be  accurately  situated  in  the  limiting  lines,  and  though  not  strictly  the 
first  and  last  points  of  these  lines,  they  will  be  very  nearly  so. 

IV.    DETERMINATION  OF  THE   PLACE  WHERE  A  GIVES  PHASE  WILL  APPEAR  BOTH  AI 
SUNRISE  AND  SUNSET. 

We  have  seen  (page  341)  that  when  the  rising  and  setting  lines  of  a  phase  ex- 
tend throughout  the  eclipse,  they  will  compose  the  figure  of  an  8  much  distorted. 
The  point  of  intersection  or  nodus  is  a  place  where  the  phase  will  be  seen  to  begin 
and  end  in  the  horizon ;  that  is,  it  will  either  commence  at  sunrise  and  end  at 
sunset,  or  commence  at  sunset  and  end  at  sunrise.  At  the  time  of  the  middle  of 
the  eclipse,  the  sun  will  therefore  be  very  nearly  on  the  meridian :  if  diff.  dec,  and 
S  are  of  the  same  sign,  it  will  be  midnight,  because  the  pole  of  the  earth  will  have 
the  zenith  and  sun  on  opposite  sides  of  it ;  but  when  those  values  are  of  different 
signs,  it  will  be  noon  at  the  place,  for  then  the  zenith  and  sun  will  be  both  on  the 
same  side  of  the  pole.  If  r  denote  the  semi-duration  of  the  eclipse,  which  begins 

and  ends  with  the  given  phase,  r  —  will   express  the  semi-diurnal  arc  of  the 

sun;  and.-,  —tan  I  tan  5  =  cos  (Tjj)   =  cos    (r   .    15°),   which   being  nearly 

unity,  we  must  have  f  ~  X  or  Z  nearly  =  90°.     Consequently  for  the  values  c 

dtt    d  v 

U*  V'  d7'  d~t'  at  the  t"ne  °f  the  micl(*le  of  the  ecliPse»  which  will  be  either  noon 
or  midnight,  we  may  assume  sin  Z  =  unity,  and  Jf=0°  or  180°.  So  we  get, 
from  the  equations  (1)  and  (2),  page  344-5, 

«  =  -  (*)  ±  P',  f  =  (y), 


Let  ft  denote  the  hourly  motion  on  the  apparent  relative  orbit,  and  i'  the  incli 
nation  with  a  parallel  of  declination  ;  then 

, d  v        .     , du 

~ It'  *  81  ~~  ~dt  ' 

or, 

HBmi'=7)i  )  , 

n  cos  •'  =  a,  cos  D  ±  [9.41796]  Pf  sin  i   \ 

The  condition  for  the  greatest  phase  is  u  - — \-  v  —  =  0,  or  u  sin  »'  —  v  cos  •'=  0 

that  is, 

[  —  (x)±  P1]  sin  ,'  —  (y)  cos  <'  =  0. 

If  t  denote  the  interval  past  the  time  of  the  true  conjunction,  we  shall  have 

(ar)  =  diff.  dec.  -f  t  A  and  (y)  =  talcos&; 
.:  [ —  diff.  dec.  ±  P']  sin  i'  —  t  [Di  sin  «'  -f  a!  cos  D  cos  «']  =s  0 ; 

/    A  \  /   -Di   \ 

or,  since  Di  =  I  — I  sin  i,  «;  cos  D  =  (  -^ I  cos  i, 

\  sm  i  /  \  sm  »  / 

[—diff.  dec.  ±  P'J  sin  ('  —  t  -^-  cos  (»'  ~  t)  =a 


350  SPHERICAL    ASTRONOMY. 


Assume  ^-  _ 

COS  (t    ~  t) 

k  sin  t'  sin  t 

and  then  t  =  -  K  -  ,  or  since  A  =  /*  sin  t', 
D\ 

t  =  ^!ln_L  or  tin  seconds  =  [3.55630]  ^-1  .....     (3) 

When  diff.  dec.  is  negative,  M  =  180°,  and  the  lower  sign  of  P'  must  be  used  ; 
or,  as  a  general  rule,  P'  must  be  used  with  the  same  sign  as  that  of  diff.  dec.,  and, 
since  I  nearly  =  90°  ~  5,  we  can  previously  correct  the  horizontal  parallax  for  the 
place  by  reducing  it  to  a  latitude  equal  to  the  complement  of  J.  The  value  of  t  be- 
ing found,  we  shall  have  at  the  place 


when  diff.  dec.  and  t>  have  j  ^s**™*  \  signs,  app,  time  of  true  6  =  j  ^  [  —  <  (4) 


which  compared  with  the  Greenwich  apparent  time  of  the  true  conjunction  will 
show  the  longitude  of  the  place. 
For  the  values  of  u  and  v  we  have 

u  =  ( —  diff.  dec.  ±  P1)  —  t  DI  =  k  cos  (t'  ~  t)  —  k  sin  i'  sin  t  =  k  cos  t'  cos  t, 
v  =  t  at  cos  D  =  t  DI  cot  i  =  k  sin  t'  cos  <. 

Let  ri  be  the  nearest  apparent  approach  of  the  centres  ;  and  the  semi-duration 
r  will  be  determined  by  the  equations 

v  ri  A'  sin  u>  sin  i' 

n  •=•  — — r,         cos  o>  =  — ;,        T  = =• . 

Bin «'  A  A 

and  thence  the  latitude  by  the  equation, 


tan  6 
Or,  using  the  above  value  of  v, 

k  cos  t  A'  sin  »  .  cos  (T  .  15°) 

— '        r  =  ~V~'         tan'=±— n—    '    •    (6) 
the  latitude  being  of  the  same  name  as  diff.  dec. 

The  middle  of  the  eclipse  will  not  have  the  sun  in  the  horizon,  except  k  cos  t  =  A  , 
T  =  0,  /  =  90°  ~  £,  and  therefore,  unless  these  particular  values  should  happen, 
the  place  will  not  range  exactly  in  the  line  whereon  the  middle  of  the  eclipse  ia 
seen  at  sunrise  or  sunset ;  this  line,  which  we  are  about  to  notice,  will  pass  the 
intersection  at  a  higher  latitude,  and  will  form  a  very  small  triangle  with  the 
rising  and  setting  limits. 

V  PLACES  WHICH  WILL  HAVE  THE  MIDDLE  OF  THE  ECLIPSE  WITH  THE  SUN 
IN  THE  HORIZON. 

In  the  first  place,  we  shall  suppose  the  inclination  of  the  apparent  orbit  to  be 
the  same  as  that  of  the  true.  The  condition  for  the  middle  of  the  eclipse  will  then 
be  simply  to  have  the  apparent  place  of  the  moon  somewhere  on  the  line  of  near- 
est approach. 

On  both  sides  of  S  take  Sm=Sm'  =s  +  or,  andm,  m'  will  be  the  limits  be- 
tween which  the  apparent  place  must  be,  in  order  that  an  eclipse  may  result.  On 
the  orbit  make  M'  m'  =  P'.  Then  if  iri  falls  between  8  and  n,  this  will  be  the 
first  position  in  which  the  ecliose  can  take  place.  But,  if  m'  falls  beyond  the 


APPENDIX    XI. 


351 


point  «,,  the  first  position  of  the  moon  F'C-  8- 

will  be  at  Mt  where  Mn  —  P' ;  and 
in  this  case,  for  each  position  between 
M  and  M'  there  will  evidently  be  a 
position  of  m'  on  both  sides  of  the  or- 
bit, and  consequently  two  correspond- 
ing places  on  the  earth;  when  the 
moon  arrives  at  M'  the  remote  point  m'  will  be  receding  from  S,  and  will  at  that 
time  get  beyond  the  limit  of  an  eclipse,  so  that  the  other  point  m'  only  will  pro- 
duce an  eclipse  under  the  assigned  conditions. 

Again,  when  m  n  is  greater  than  P',  it  is  evident  that  these  limits  will  continue 
throughout  the  whole  duration  of  M'M'  or  MM.  When  m  n  is  less  than  P't  by 
making  m  M"  =  P'  the  limits  for  an  eclipse  will  end  at  the  point  M",  and  it  will 
be  impossible  throughout  the  duration  of  M"  M".  These  two  cases  are  the  same 
as  those  distinguished  in  the  rising  and  setting  limits,  page  340,  s  +  a  being  the 
value  of  A '. 

To  determine  the  times  between  which  these  phases  are  possible,  or  the  semi- 
durations  answering  to  the  positions  M,  M1,  M",  we  shall  in  each  instance  denote 
the  angle  M  m  n  by  the  character  o>,  and  the  following  equations  will  be  readily 
deduced. 

(1)  When  n  <  P1  —  (s  +  ff), 


n  + 


.    .    .    .    (1) 


w2  >  90°  when  diff.  dec.  is  negative. 

These  semi-durations  will  give  two  times  of  beginning  and  ending;  the  one  an- 
swering to  the  point  M  and  the  other  to  the  point  M".  The  middle  of  an  eclipse 
in  the  horizon  will  take  place  from  the  first  beginning  to  the  second  beginning, 
and  from  the  second  ending  to  the  first  ending. 

The  places  will  be  determined  by  producing  m  M  to  a  distance  of  90°  from  m. 
If  a  great  circle  be  drawn  through  S,  so  as  to  be  at  this  point  parallel  to  m  M,  it 
will  evidently  intersect  the  former  at  a  distance  of  90°  and  determine  the  same 
place.  We  shall  therefore,  in  supposing  the  places  to  be  determined  in  this  man- 
ner, have  the  following  formulae  : 

First  place  of  beginning,  wj  =  90°, 


sin  I  =  —  sin  «  cos  6, 


tan  h  = 


cot  t 
sin  S 


It  must  be  taken  in  the  2d  semicircle,  or  between  0°  and  —  180°    , 
First  place  of  ending, 

Change  the  name  of  the  latitude  of  the  place  of  beginning,  and  to  the  hour  angle  h 
apply  ±  180°.    The  results  will  determine  the  place  of  ending. 
•  Second  place  of  beginning, 

tan  a 

sin  <5 


sin  I  =  cos  a  cos  $, 
Second  place  of  ending, 

sin  /  =r  cos  b  cos  t, 


h  = 


tan  h  =  — 


tan  b 
sin  6 


. 

(8) 


352  SPHERICAL    ASTRONOMY. 

The  second  places  of  beginning  and  ending  will  be  two  of  the  extreme  points  of 
the  lines  traced  on  the  earth.     The  other  two  extremes  may  be  determined  by 

computing  cos  w  =  -  -^  -  ,  and  proceeding  as  before,  observing  that  n  must 

be  considered  positive,  and  «  >  90°  when  diif.  dec.  is  positive.  These  four  extiviut 
points  are  the  same  as  those  of  the  northern  and  southern  limits,  the  phase  being 
simply  external  contact. 

2)  When  n  >  f  —  (s  +  ff)  and  <  s  +  ff, 

The  places  will  be  determinable  throughout  the  whole  of  the  first  ^   .    .  (4) 
duration  found  as  above. 

(3)  When  n  >  s+  ff, 


n  —  (s  +  ff) 
cos  w  =  -  ^—  -  Z, 
Jr 


n  must  here  be  considered  a  positive  quantity,  and  w  will  be  >  90C 

when  diff.  dec.  is  negative. 
The  phase  will  continue  throughout  the  whole  duration,  and  the  ex 


treme  places  may  be  computed  from  this  value  of  w  according 


.  (5) 


to  the  equations  (3). 

Having  found  the  limits  between  which  the  phase  is  possible,  the  places  for  any 
intermediate  times  may  be  determined  thus, 

t  denoting  the  time  from  the  middle, 


(D  >  90°  when  diff.  dec.  is  negative, 
and  the  places  by  the  equations  (3). 

If  n  <  s  -f-  ff,  suppose  n  to  be  positive,  and  compute 


=  ( )  sin  u. 


Then  for  times,  without  the  limits  of  this  duration,  we  may  determine  four 
places  ;  two  with  w  <  90°  and  two  with  o>  >  90°,  which  will  all  fulfil  the  necessary 
conditions. 

The  preceding  results  have  been  derived  on  the  assumption  of  t'  =  t.  They 
will  be  sufficiently  approximate  for  a  general  drawing  of  the  lines  on  a  map,  and 
more  particularly  as  these  phenomena  cannot  be  subject  to  minute  observation. 
When,  however,  from  local  circumstances  or  otherwise,  greater  accuracy  is  wanted, 
we  must  use  the  proper  value  of  t'  and  the  relative  horizontal  parallax  reduced 
to  the  latitude  thus  determined.  Since  Z  =  90°,  the  condition  for  the  middle  of 
the  eclipse,  according  to  the  equation  (4)  page  346,  is  «'  —  v  =  0  or  i'  =  v.  Let  the 
figure  at  page  344  represent  the  positions  which  answer  to  the  particulars  of  the 
present  case.  Then  as  M  m  =  Mm'  =  P',  the  Z  Mmm'  •=  /  Mm'  m.  Denote 
this  angle  by  6;  the  angles  Nm  M,  Nm  M  by  M,  M1  ;  and  we  shall  have 


M=o—v, 

180°—  S—  »,       /. 

8+v  —  9,  £  SMm!  =180°—  (8  +  v  +  0). 


APPENDIX    XI. 


353 


With  the  triangles  MSm,  M Sm',  we  hence  find 
sin  0  =  -p  sin  (S  +  v) ; 


—  pi  8in  O8  +  y  —  g)  <; 

sin  (S  +  v) 

wlack,  for  computation,  may  be  thus  arranged 
sin  (8  +  v) 


__       sin  (8  +  v  +  9) 


9= -pi -i  sin  0  =  $r.  A; 

J  to  be  +  or  —  but  less  than  90°  ; 


Sm'  = 


9 

,  _  sin  (S  +  v  +  0) 


.     (6) 


The  points  m,  m',  may  in  some  cases  be  both  on  the  same  side  of  S,  and  the 
value  of  Srn  is  only  necessary  to  indicate  whether  any  portion  of  the  sun  id 
eclipsed  or  not.  To  have  an  eclipse,  Sm,  taken  as  a  positive  quantity,  must  be 
less  than  s  -f  <r,  and  we  must  only  determine  a  place  from  the  angle  M  when  the 
corresponding  value  of  S  m  is  within  this  limit.  If  8  m,  S  m',  taken  as  positive 
quantities,  are  both  greater  than  «  +  <r,  the  middle  of  an  eclipse  cannot  be  seen 
on  the  earth  under  the  assumed  conditions;  on  the  contrary,  if  Sm,  S  m'  so  taken 
are  both  less  than  s  +  r,  the  angles  M,  M'  may  both  be  vsed,  and  consequently  two 
places  will  be  determined.  In  each  case,  similarly  to  (3),  we  adopt  the  formulae 


sin  I  =  cos  M  cos  a, 


tan  h 


—  ^.n_^  \ 
sin  o    ) 


(7) 


VI  CENTRAL  LINE. 

The  places  which  in  succession  see  a  central  eclipse  are  evidently  determined 
by  producing  S  M  to  a  distance  Z  from  8,  so  that 


(1) 


for  then  the  relative  parallax  P'  will  bring  the  centres  to  a  coincidence.  To  de- 
termine the  position  of  the  place  on  the  earth  for  any  given  time,  we  have  in  the 
triangle  N  S  Z,  thus  formed,  NS  =  W°  —  S,  ^NSZ  =  S,  SZ  =  Z,  and  hence 
the  following  formulae : 

tan  0  =  tan  Z  cos  S, 
6  to  be  +  or  —  and  less  than  90° ; 


tan  /  =  tan  (0  -f-  <J)  cos  h, 


cos  (0  -f  8) 
h  to  be  in  the  same  semicircle  wuh  S; 
sin  0  sin  Z  cos  8 


check 


cos  (0  -f-  f>)       cos  h  cos  I 


(*) 


In  the  course  of  the  general  central  eclipse,  one  of  the  places  on  the  earth  will 
have  the  central  eclipse  at  noon.     At  this  instant  the  bodies  will  obviously  have 

23 


SPHERICAL   ASTRONOMY. 

true  as  well  as  apparent  conjunction  in  right  ascension,  and  .*.  A  =  diff.  dec.  and 
S  =  0.     This  place  is  hence  determined  thus : 

_,      diff.  dec.  11/7 

6inZ  = — ,  1  =  6 +  Z, 

(3) 

Z  to  have  the  same  sign  as  diff.  dec. 

App.  time  of  true  6  =  west  long,  of  place, 

These  equations  (1),  (2),  (3),  involve  the  horizontal  parallax  P',  answering  to  a 
mean  latitude  of  45°,  which  will  be  sufficiently  near  for  ordinary  purposes.  Where 
an  accurate  result  is  wanted,  the  calculation  must  be  repeated  with  the  use  of  the 
equatorial  relative  parallax  properly  reduced  to  the  latitude  thus  determined. 

The  first  and  last  places  on  the  earth  which  see  a  central  eclipse,  are  to  be 
found  by  the  formulae  at  pages  338-40. 

The  preceding  discussions  comprise  all  that  is  necessary  for  the  calculation  of 
the  lines  which  are  shown  in  the  ma-ps  now  inserted  in  the  Nautical  Almanac,  and 
which  are  quite  sufficient  to  indicate  the  general  character  of  the  eclipse  that  may 
be  expected  for  any  particular  place.  We  might  now  proceed  to  show  the  appli- 
cation of  these  equations  in  the  resolution  of  innumerable  other  curious  and  in- 
teresting problems;  but  such  a  field  of  speculation  would  not  conform  with  the 
object  of  this  paper,  and  may  the  more  willingly  be  abandoned  on  the  considera- 
tion that  the  means  of  solution  may,  in  most  cases,  be  readily  elicited  from  the 
equations  already  established.  The  following  classification  of  these  equations  will 
be  found  to  exhibit,  in  a  comprehensive  form,  all  that  will  be  requisite  to  direct 
and  facilitate  the  operations  of  the  calculator,  and  relieve  the  mind  from  any  un- 
necessary reference  or  consideration. 


NOTATION. 

D  =  the  D  's  true  declination ; 
d  =  the  O's  true  declination; 
a  =  the  true  difference  of  right  ascension  in  are, 

or  ])  '&  right  ascension  —  O's  right  ascension ; 
D1  =  the  D  's  relative  motion  in  declination, 

or  J)  's  motion  in  declination  —  ©'s  motion  in  declination, 
ai  =  the  D  's  relative  motion  in  right  ascension, 

or  the  motion  of  the  D  —  that  of  the  0  ; 

Diff.  dec.  =  the  true  difference  of  declination  at  c5  in  right  ascension, 
viz.,  D 's  declination  —  ©'s  declination,  at  that  time; 
P  =  the  D  's  equatorial  horizontal  parallax  ; 
TT  =  the  ©'s  equatorial  horizontal  parallax; 
P'  =  [9.99929]  (P-»); 
s  =  the  D 's  true  semi-diameter ; 
»=  the  ©'s  true  semi-diameter; 
A  =  the  true  distance  of  the  centres ; 
Z>',  a',  s',  A',  the  apparent  values  of  J),  a,  s,  A; 

«  =  the  angle  under  A  and  n:  in  all  cases  this  angle  is  to  be  tak?n  pos- 
itively, and  between  0°  and  180°. 


APPENDIX   XI.  355 


I. BEGINNING  AND  ENDING  OF  A  PHASE  ON  THE  EARTH. 

1.  (D,  A  and  ai  at  6); 

tan  t  = -  ;  n  =  diff.  dec.  X  cos  t ; 

ai  cos  D 

i  of  the  same  sign  as  DI  ; 

n  of  the  same  sign  as  diff.  dec. 


n  sin  t  [3.556301 

c  =  --   —  -    -J; 


sin  i  to  be  found  by  combining  the  preceding  values  of  cos  t  and  tan  i; 
sign  of  t  to  be  determined  by  diff.  dec.  X  -Di. 

Time  of  middle  =  time  of  <3  —  t ; 
C  partial   "j 

T-,        I  central 

For  4-  }-  eclipse,  A  = 

I  total 

\  annular  J 

n 
cos  w  =  — ;  T  =  c  tan  w. 


«=(-«)-„;  ft.=  (_0  +  c* 

4.  Place  of  beginning,  (S  at  <5  ) ; 

.     .  tan  a 

sm  /  =  cos  a  cos  S :  tan  h  = : — r  ; 

sin  i 

H=  apparent  Greenwich  time  of  beginning; 

longitude  east  =  h  —  H\ 
h  to  be  in  the  same  semicircle  with  a. 

5.  Place  of  ending,  (S  at  c5 ) ; 

sin  /  =  cos  6  cos  &  •  tan  h  = : — -  : 

Bint  ' 

JET=  apparent  Greenwich  time  of  ending; 

longitude  east  =  //  —  H\ 
h  to  be  in  the  same  semicircle  with  6. 

6.  For  more  accurate  calculations,  reduce  the  true  relative  horizontal  parallax, 
oy  means  of  the  table  at  p.  387,  to  the  latitudes  so  determined,  and  recompute. 


H. KISING  AND  SETTING  LINES. 

For  partial  eclipse,  A '  =  «  -f-  «r. 

7.  When  n  >  P'  —  A '. 

These  limits  will  extend  throughout  the  entire  duration  of  the  general  eclipse, 
and  form  the  distorted  figure  of  an  8,  the  first  and  last  points  being  the  places  ol 
beginning  and  ending  on  the  earth. 


350  SPHERICAL    ASTRONOMY. 

8.  When  n  <  P  -  A  '. 

"With  P'  —  A  ',  instead  of  A  ',  compute  as  for  the  times  of  beginning  at  1  ending 
on  the  earth  ;  and  let  these  times  be  t\t  fa.     Then 

the  risings  j  ^in  j  at  j  Partial  beginninS> 
in  which  interval  the  first  oval  will  be  completed  : 


in  which  interval  the  second  oval  will  be  completed. 

The  limiting  places  at  the  times  t\,  fa,  are  to  be  found  in  the  same  manner  aa 
the  places  of  beginning  and  ending  of  a  phase  on  the  earth. 

9.  Places  for  any  times  within  the  limits  : 

Prepare  the  constants,  p  —  -  -  -  ,     q  =  —  —  -  , 

and  let  t  be  the  time  from  the  middle  of  the  general  eclipse  ; 

t  n 

tan  w  =  —  :  A  =  -  ; 

c  cos  m 

w  >  90°  when  n  is  —  . 

10.  8=(-t)¥». 

the 


"•  A  A  ~YT     ZT 

p  )    f  n 1 

,2         p)    V        2/ 


.    m 
sin  —  = , 

2       r  JT  .  A 

771 

—  to  be  less  than  90°  and  positive 

12.  Place  following, 

sin  /  =  cos  (8  —  m)  cos  S ;  tan  h  = -. — ; — '; 

sin  t 

H=  apparent  Greenwich  time; 

longitude  east  =  h  —  H ; 
h  to  be  in  the  same  semicircle  with  8  —  m. 

13.  Place  advancing, 

tan  (S  -4-  m) 
sin  /  =  cos  (8  -|-  wi)  cos  ^ ;  tan  h  = ^~"A — ' ' 

longitude  east  =  h  —  H; 
h  to  be  in  the  same  semicircle  with  8  +  m. 

14.  For  a  more  accurate  determination,  find  the  values  of  D,  t,  a  for  the  given 
time,  a*d  P'  =  p  (P  —  w)  for  the  latitude ;  thence 

(D)  =  J)  +  (a  corr.  from  table,  p.  342) ; 


APPENDIX    XI.  357 


.    wt 

sin  — 


* 


The  quadrant  of  $  to  be  determined  by  (z),  (y),  as  co-ordinates. 
With  these  values  of  S,  m,  compute  the  places  by  Nos.  12  and  18. 


When  n  >  P'  —  A  '. 

15    Find  P'  =p  (P  —  ir).  for  a  latitude  equal  to  the  complement  of  $  at  rf. 
/.sin*'  =  .01, 
n  cos  t'  =«i  cos  D  ±  [9.41796]  P'  sin  S, 


.  t  .H  mmds  = 

COS  (*'  ~  «) 


1  6.  At  the  place, 
When  diff.  dec.  and  t  have  j  J^^6  |  signs,  »PP-  time  of  true  6  =  j  ^h  [  -  *, 


which,  compared  with  the  Greenwich  apparent  time  of  the  true  <3,  will  determine 
khe  longitude  of  the  place. 

17.                                      k  cos  i                                A'  sin  <» 
cos«  =  — -T-,  T  = , 


I  to  be  of  the  same  name  as  diff.  dec. 

IV.     PLACES  WHERE  THE  MIDDLE   OF   THE   ECLIPSE  IS   SEEN  WITH   THE 
SUN  IN  THE  HORIZON. 

1 8.   When  n  <  P'  —  («  -J-  »),  compute 

c  P'                             n±(s  +  <i) 
TI  = ,  COS  wa  = ^ -,  ra  = 

using  s  -f-  9  with  a  sign  the  same  as  that  of  n. 

These  semi-durations  give  two  times  of  beginning  and  ending  ;  the  phenomenon 
will  take  place  on  the  earth  between  the  times  of  beginning  and  between  the  times 
of  ending. 

The  places  of  first  and  last  appearance  on  the  earth  to  be  determied  thus  : 

For  first  appearance, 

sin  I  =  —  sin  t  cos  &,        tan  h  =  —  — : . 

•in  a 


358  SPHERICAL  ASTRONOMY. 

For  last  appearance,  change  the  name  of  the  latitude  of  the  former  place,  and 
to  the  hour  angle  h  apply  ±  180°. 
For  the  extreme  points  compute  also 

n  T  (s  +  <r)  /  e  P'\    . 

cos  03  = ~ -,  TS  =  \—^~)  8m  "a '» 

using  s  -j-  <r  with  a  sign  contrary  to  that  of  n. 

Then  with  the  values  of  o>2,  u3,  proceed  as  for  the  beginning  and  ending  ot  h 
phase  on  the  earth. 

When  diff  dec.  is  +,  j  ^a  j.  gives  points  meeting  j  ™§l™  [  limit. 

When  diff.  dec.  is  -,  j  £  [  gives  points  meeting  j  ~°^  |  limit. 
The  eclipse  will  be  visible  on  both  sides  of  the  equator. 


19.  When  n  >  P'  —  («  +  ff)  an<l  <  s  +  ff»  compute 

=  — 

», 

The  phenomenon  will  continue  throughout  the  whole  of  the  duration  so  found. 
The  two  extreme  points  will  be  determined  as  above  with  the  angle  w3. 
The  places  of  first  and  last  appearance  also  as  above. 

20    When  n  >  *  -\-  a,  compute  «3,  rs,  as  above. 

The  phenomenon  will  continue  throughout  the  whole  duration,  and  the  extreme 
places  will  be  determined  by  proceeding  with  this  value  of  w  as  for  the  beginning 
and  ending  of  a  phase. 

These  places  will  in  this  case  be  also  those  ol  first  and  last  appearance. 

21.  Places  for  any  time  within  the  limits  : 

Let  t  be  the  time  from  the  middle,  and  compute 


If  n  <  s  +  8,  this  u  may  be  taken  both  greater  and  less  than  90°  when  t  is 
greater  than  r3  before  found  ;  and  then  four  places  will  be  determined.  In  all 
other  cases  whatever  w  must  be  >  90°  when  diff.  dec.  is  negative. 

The  places  to  be  determined  by  proceeding  with  w  as  for  the  beginning  and  end- 
ing of  a  phase. 

22.  For  a  more  accurate  determination  at  any  time  : 

Find  P'  =  f  (P  —  IT)  for  the  latitude  before  found. 
Find  (a?),  (y),  8,  and  A,  as  in  No.  14. 
For  the  time  of  (5  form  the  constants 

(A)  =  [0.58204]  a,  cos  D,  (B)  =  [0.58204]  A- 

Compute  v  from  the  equations, 


P'  cos  a 


APPENDIX    XI  359 


23.  Then  sin  (S+v) 

9  —  —  -T—^  •>  em  6=g  .  A, 


9 
0  to  be  +  or  —  but  less  than  90°. 

24.  tan  M 

sin  I  =  cos  M  cos  &,  tan  h  =  --  :  —  -. 

ami 

If  *,  «',  be  both  less  than  s  +  <r,  the  angles  M,  M',  may  be  both  used  in  these 
equations,  and  two  places  determined.  If  one  of  the  quantities  «,  «',  be  greater 
than  s  +  <r,  the  corresponding  M  will  be  excluded,  and  only  one  place  determined 
with  the  other  value.  If  *,  «',  be  both  greater  than  s  +  a,  both  computations 
will  be  excluded,  and  the  assumed  time  will  be  without  the  limits  of  the  appear- 
ance on  the  earth. 


V.    NORTHERN  AND  SOUTHERN  LIMITS  FOR  ANY  PHASE. 

C  Partial  J  C(s  +  6")  +  <r, 

For  •? Total      >  appearance,  A'  =  <(s  +  6")  —  o, 
(Annular)  (  ff  —  (s  +  6"). 

6"  is  added  as  a  mean  augmentation  of  s. 

25.  When  n  <  P'  —  A '  both  limits  will  have  place. 

When  n  >  P'  —  A '  only  one  limit  will  have  place,  viz. : 


26.  First  and  last  points  or  places  of  entrance  and  departure : 

n±  A'                            /cP'\ 
cosw  =  — =r, — ,  r=| Isinw; 


TepaTre  j  =  time  of  midd"  |  ~  \  '• 

Places  of  entrance  and  departure  determined  as  in  Nos.  4  and  6,  for  the  begin- 
ning and  ending  of  a  phase,  using  a  =  ( —  i)  —  w  and  6  =  ( —  i)  +  w. 

For  the  appearance  of  external  contact  these  determinations  are  included  ic 
No.  18,  and  therefore  need  not  be  repeated  for  these  limits. 

27.  Places  for  any  times  within  the  limits : 

Prepare  the  following  constants,  using  6  at  (5, 

A'  sin  i 
u  =  A   cos  «,  D  =  o  T  w,  a  =  ± 


cos  D'  ' 

„  n  n  ±  A'         , 

E  =  — -,  cos  w  = =-,: —  as  above  ; 

c  (n  ±  A')  P 


360  SPHERICAL    ASTRONOMY. 

28.  Let  t  be  the  time  from  the  middle  of  the  general  eclipse, 

Jf=(-OTa>'; 


29. 
tan  (A  -  a')  = 


sin  9 


cos  (f  +  D') 
check     .     .     . 


tan  e  =  tan  Z  cos  Jf, 

tan  Jf,  tan  /  =  tan  (e  +  D')  cos  (h  —  a 

sin  0  sin  Z  <:os  M 


cos  (6  +  J)1)       cos  (A  — a')  cos  I' 
<  90°,  and  same  sign  as  cos  M ;  and  A  —  a  to  be  in  the  same  semicircle  with  M. 

30.  For  a  more  accurate  determination  at  any  time, 

Find  P'  =  p  (P  —  *)  for  the  latitude  before  found. 

Also,  with  Z  find  the  augmented  semi-diameter  s'=s  +  augmentation,  from 
the  table  annexed. 


Z 

Augmentation  of  the  D's  Semi-diameter. 
Argument  :  True  Zenith  Distance  Z. 

For  P 
=  54' 

Var.  for 

10' 

in  P. 

Z 

ForP 

=  54' 

Var.   for 

10' 

in  P. 

Z 

ForP 
=  54' 

Var.  for 

10' 

in  P. 

o 

// 

ii 

0 

n 

(/ 

o 

/, 

„ 

o 

i4-o 

5.7 

3o 

I2«  I 

4-9 

60 

6-9 

2.9 

I 

i4-o 

5-7 

3i 

12-0 

4-8 

61 

6.7 

2.8 

2 

i4-o 

5.7 

32 

11-9 

4-8 

62 

6.5 

2.7 

3 

i4-o 

5-7 

33 

II-  7 

4-7 

63 

6.2 

2-6 

4 

i4«o 

5.7 

34 

ii.  6 

4.7 

64 

6.0 

2.5 

5 

i3.9 

5.7 

35 

n.  5 

4.7 

65 

5.8 

2.4 

6 

i3.9 

5-7 

36 

ii.3 

4-6 

66 

5-6 

2.3 

7 

i3.b 

5.7 

37 

II.  2 

4-6 

67 

5.4 

2-2 

8 

i3.8 

5.7 

38 

II  -0 

4-5 

68 

5-2 

2-1 

9 

i3.8 

5-7 

39 

10-8 

4-4 

69 

4-9 

2-O 

10 

i3.8 

5-6 

4o 

10.7 

4.4 

70 

4.7 

•9 

ii 

i3.7 

5-6 

4i 

io-5 

4-3 

71 

4-5 

•  8 

12 

i3.7 

5-6 

42 

10-3 

4-3  . 

72 

4-2 

•7 

i3 

i3.6 

5-6 

43 

10  -2 

4-2 

73 

4-o 

•  6 

i4 

i3.6 

5-5 

44 

IO-O 

4-i 

74 

3-8 

•  5 

i5 

i3.5 

5-5 

45 

9.8 

4*i 

7^ 

3-5 

•4 

16 

i3.4 

5-5 

46 

9.7 

4-o 

76 

3-3 

•  3 

i? 

i3-4 

5-4 

47 

9.5 

3.9 

77 

3-1    ' 

•2 

18 

i3.3 

5-4 

48 

9'3 

3.0 

78 

a-8 

I  -I 

19 

13.2 

5-4 

49 

9.2 

3-8 

79 

2-6 

I  «I 

20 

i3.i 

5-4 

5o 

9-0 

3.7 

80 

2-4 

I  'O 

21 

i3.o 

5-4 

5i 

8-8 

3-6 

81 

2«I 

0-9 

22 

12.9 

5-3 

52 

8-6 

3-5 

82 

1-9 

0.8 

23 

12.8 

5-3 

53 

8-4 

3-4 

83 

1-7 

0.7 

24 

12.7 

5.3 

54 

8-2 

3.3 

84 

1,4 

0-6 

25 

12.6 

5-2 

55 

8-0 

3-2 

85 

1-2 

0-5 

26 

12.5 

5-i 

56 

7.'8 

3.2 

86 

t'O 

o-4 

27 

12.4 

5-i 

57 

7-5 

3.i 

87 

0-7 

o.3 

28 

12.3 

5-0 

58 

7.3 

3-1 

88 

0-5 

0-2 

29 

12.2 

4.9 

59 

7-1 

3.o 

89 

o-3 

0-1 

3o 

12.  1 

4.9 

60 

6-9 

2-9 

90 

0-0 

o«o 

APPENDIX   XI.  36J 


Then,  f  Partial   J  r  s1  +  a, 

For  j Total      V  phase,  A'  =  J  «'  —  *, 

f  Annular  )  t  a  —  a? 


81.  For  the  time  of  d  form  the  constants, 

(A)  =  [0.58204]  ai  cos  D,  (B)  =  [0.58204] 

Find  the  values  of  D,  S,  a,  for  the  given  time. 

(D)  =  D  +  (a  corr.  from  table,  page  342). 
(»)  =  (/>) -4,  (y)  =  «cos(J9), 


•  »     www    §r    _. 

P'  COS  5 

82.  (Z  from  the  first  computation), 
sin  0  =  /4/ 


cos  Z  tan  » 

.  -  ,  tan  t  =  - 

2  A  cos  v  cos  2  ^ 

u  =  A  '  cos  t',  D'  =  S  T  M, 

v  =  A'  sin  t',  o'  =  ± 


—  , 
cos  D 
—  a')  corr. 


Remaining  computation  the  same  as  in  No.  29. 


VI.     CENTRAL   LINE. 

38.  The  computation  of  the  limiting  times  and  places  is  comprehended  undei 
the  head,  "  Beginning  and  Ending  of  a  Phase  on  the  Earth." 

34.  Places  for3  any  times  within  the  limits: 

t  =  the  time  from  the  middle. 

t  n 

tan  w  =  -,  A  =  -  k 

c  cos  * 

w  >  90°  when  n  is  negative. 
85  £  =  —i)  T«; 


86  (S  at  6). 

sin  Z  =  -^,  tan  0  =  tan  Z  cos  8, 

tan  h  =  —  T^X^:  tan  8,  tan  /  =  tan  (8  -f-  b)  cos  A, 

cos  (6  ~r  i) 


362  SPHERICAL    ASTRONOMY. 

sin  6       _  sin  Z  cos  S 
'  cos  (0  +  i)       cos  h  cos  I  ' 

8,  same  sign  as  cos  S,  and  less  than  90°: 
A,  same  semicircle  with  S. 

87.  For  a  more  accurate  determination  at  any  time,  find  P't  8,  A,  as  in  No.  14, 
and  proceed  again  with  these  as  in  No.  36. 

88.  Place  where  the  eclipse  will  be  central  at  noon  : 

(*•*<$)• 


Apparent  Greenwich  time  of  true  (5  =  longitude  W. 
Z  <  90°  and  same  sign  as  diff.  dec. 

39.  For  a  more  accurate  determination,  find  the  horizontal  parallax  for  the  lati- 
tude, and  with  it  repeat  the  operation. 

[All  latitudes  in  the  preceding  formulae  are  to  be  recognized  as  geocentric,  and 
will  therefore  need  reducing  by  the  table  at  page  336.] 

Examples. 

For  an  elucidation  of  the  practical  application  of  the  preceding  formulae,  we  shall 
take  the  solar  eclipse  of  May  15,  1836.  At  the  time  of  new  moon,  viz.  2*"  7m-o. 
the  moon's  latitude  0  is  a5'  43",  which  being  less  than  i°  a3'  17'',  the  eclipse  is 
certain.  (See  the  limits  at  page  333.)  The  elements  of  this  eclipse,  as  related  to 

the  equator,  are 

d.      h.  m.  s. 

Greenwich  mean  time  of  <3  in  R.  A.   .      .      .  May  i5     2  21  22-9 

D  's  declination  .........  N.  19  25  9-8 

©'s  declination  .........  N.  18  67  58-8 

D  's  hourly  motion  in  R  A       .....  3o  8  •  3 

©'s  hourly  motion  in  R.  A  ......  2  28  •  2 

D  's  hourly  motion  in  declination   ....  .              N.     9  68.7 

©'s  hourly  motion  in  declination         .     .     *  N.  35  «i 

D  's  equatorial  horizontal  parallax      ...  54  23  •  9 

©'s  equatorial  horizontal  parallax      ...  8-5 

I>  's  true  semi-diameter  .......  ,                  1  4  49-5 

©'s  true  semi-diameter  .......  1  5  49-9 

from  which  we  prepare  the  following  values  : 


Q 


D'sdec.       .      .     +J925io       D  's  H.  M.  in  R.  A.    .     3o    8 
©'s  dec.       .      .     +  18  57  59       ©'s  H.  M.  in  R.  A.    .       2  28 

Diff.  dec.     .     .  +  27  ii  o!     .     .     .     .     27  4o 


>'sH. 
©'eH. 

M.  in  dec.     .     • 
M.  in  dec. 

•*•  9 

59 
35 

D's 
©'s 

Rel. 

eq. 
eq. 

eq. 
P 

hor. 
hor. 

hor. 

par. 
par. 

par. 

.     54 
.     54" 
.    54 

24 
9 
[5 

10 

log. 
const. 

log. 

3-51255 
9'99929 
3.5u84 

Di     .'    • 

f  9 

24 

APPENDIX    XI.  363 


I.    BEGINNING   AND   ENDING   ON   THE   EAKTH. 

A  +    9'  24"    ....    2.75128  (i) 

ai          27    4<>        ....      3«22OII 

9.53117 

D  -f  19°  25'. 2     -cos  .     .     9-97456 

j  tan  .      .     9-55661  (2) 

<+I949        1  cos  .      .     9.97349  (3) 
diff.  dec.  -f-  27'  11"     ....     3-21245 

n    +  25    34       ....     3.18594 

sin  t      .     9-53oio  (2)  -f  (3) 
const.    .     3-5563o 

6-27234  (4) 


e     .     .     3.52io6  (4)  —  (i) 


t     .     .     -f.  19™  56»      .     &  tan  t  .     3-07767 

d.      h. 

6    .      i5  2  21     a3 


1 5  a     i     27     middle  of  general  eclipse 

P'     .     .     .     54'  i o''=  A  for  central  phase 
s  -f  <r  .      .      .     3o    39 

84   49   =  A  for  partial  phase 

Partial  Central. 

n     .     +  3-18594  n     .     -f  3-i8594 

A    .  3.70663  A    .  3-5n84  (log./*) 

(  cos     +  9'4793i  (  cos     -f-  9-67410 

«  .  ?a°  27'  ]       — «   01°  49'  i 

(tan     0-49999  I  tan     0-27109 

C   .      3-52106                   C   f      3-52106 
d.   h.  m.  s. d.  h.  m.  s.     

t    .          2  54  57     .     4-o2io5        T     .          i  43  17     .     3.79215 
i5    2     i  27  i5  2     i  27 

1 4  a3     6  3o  beginning  i5  o  18  10  beginning 

1 5  4  56  24  ending  i5  3  44  44  ending 

(_ «)     .      ...     —  ,9  49  (— ,)     .     .      .     _  19  46 

«...          72  27  w       .     .     .          61  49 

a       ...     —  92  16  a'       ...     —  81  38 

b       .     .     .     +  5a  38  6'       .     .     .     +  42    o 

PLACE  OF  PARTIAL  BEGINNING. 

h.  m   s. 

cos  a   .      .     —  8*59715      tana  .      .     -}-   i-4o25i       Greenwich  time  23  6  3o 

cos  S   •      '     -f-  9-97576      sin  3     .      .     -}-  9-51191       Equation     .  3  56 

sin  I    .      .     —  8-5721      tan  h  .      .     —  1-8060  time    .      .   23  id  26 


I    .     .     S.      2°    9'  A  ...     —  89°  16'  (  space        .    347 

Reduction  i  H .     .          347    37 

Latitude        S.       a     10      Longitude     W.  76    53 


364 


SPHERICAL    ASTRONOMY. 


In  the  same  manner  may  the  places  of  partial  ending  and  central  beginning  and 
ending  be  calculated,  which  will  come  out 


Partial  ending 
Central  beginning 
Central  ending 


Long.  E.  28  5 1 
Long.  W.  98  16 
Long.  E.  52  4 1 


Lat.  N.  35  1 3 
Lat.  N.  7  58 
Lat.  N.  44  5o 


II.    RISING   AND   SETTING   LIMITS. 

P' 54  10 

«  +  »=  A' 3o  39 

P'—  A' 23  3 1        p  =  n  46 

P'+  A' 84  49        ?  =  4a  25 

Since  n  >  P'  —  A',  these  limits  will  extend  throughout  the  whole  duration  01 
the  eclipse  ;  and  we  may  therefore  calculate  the  position  of  a  place  for  any  time 
between  the  Greenwich  times  i4d  23h  6ra  3os,  and  i5d  4h  56m  24s.  As  an  ex- 
ample, take  the  time  i5d  oh  3ora. 


8 
m 

S—m 

,s'+m 


Assumed  time     . 
Time  of  middle   . 


d.   h.    m.    a. 
i5  o  3o 
i5  2     I   27 

1  3i   27 3.73900 


0       ' 

.     58  5i 

c     . 
j  tan      .      . 

(  cos      .      . 
n  . 

.     3-52iot 
.     0-2182 

.     9.7140 
.     3.1859-. 

.     49  24 
.     24  42 

j  log  A       . 
}  comp. 

.     3.47191 

.     6.62809 

.     12  56  . 

2-88986 

.     17  43  . 

3-02653 

Comp.  log  P' 

.     6-48816 

0      ' 

17  0.9 

sin  -J  m    . 

2)18-93264 
.     9-46632 

cos  (S  —  m)  —  9-58648 


COS  i     . 
• 
sin  I    . 

I    .      . 
Reduction 

Latitude   . 

-»-  9-97576 

sin  &     . 
tan  h  . 

h  . 
H  . 

Longitude 

.      +  9.51191 

—  9.56224 

.     —0.86659 

S.    21°    24' 

8 

.     —  82°  i5' 
8     29 

S.    21       32 

i  .    W  90    44 

PLACE  FOLLOWING. 

h.  m.  a 

tan  (S — m)  +  o-3785o      Greenwich  time  o  3o    a 
Equation         +         3  56 

(  time      o  33  56 


ZTin 


(  space       8°  29 


APPENDIX   XI.  365 

PLACE  ADVANCING. 

uos  (S  +  m)     .      .      +  9«85225  tan  (S  +  m)     .      .     —  9.99444 

cos  3     .      .     .      .     +  9-97576  sin  i      .      .      .      .      +  9.51191 

tan  h     .      .      .      .      +  0.48253 


h     .      .      .      .     —    108  i3 
Reduction  n  H  8  20 


Latitude  .     N       42  29  Longitude  .      .      .     "Vf.   116  42 


By  taking  S  =  (  —  «)  +  w  instead  of  (  —  <)  —  w,  similar  computations  will  give 
the  places  following  and  advancing  for  the  interval  t  =  ih  3im  27"  after  the  time 
of  middle,  or  for  the  Greenwich  time  i5d  3h  32m  548.  Much  time  will  be  saved 
by  taking  the  computations  two  and  two  in  this  manner. 


in.    PLACE   WHERE   THE   RISING   AND   SETTING   LINES   INTERSECT. 

o      / 

90    o 

a  18  58 


71 


, 

—  diff.  dec. 
+  P1  .     . 


co*  i 


A' 

COB* 
sin  • 


P    • 

P  —  IT         . 

.     .         9-99872 
.     .         3-51255 

54'  5" 

.     .         3.51127 

sin  S  .     . 
const. 

.     .     +  9.51191 
.     .         9-41796 

+    4  36     

.     .     +  2-44114 

cosD      . 

3-22OII 

.     .         9-97456 

+  26    6     

.     .         3.19467 

3o  42  =  fi  cos  «'     . 
ft  sin  t'     . 

(tan 

.     .     +  3-26529 
.     .     +  2.75128 

+  o«485oQ 

•     '7°  ''    |C09    .      .      . 

.     .     +  9.98056 

10  4o       u 

.     .     +  3.28473 

.       248 
—  27'  n" 
+  54    5 
+  26  54     

+  3  .20700 

+  3.20842  

9-99948 
+  3-2o84a 

+  9-97349      sin*  .      .      . 

+  9«53oio 

+  3-18191 
3-26458 

3-5563o 
f  6-29482 

9.91733      \OSf 

+  3*oiooo 

366  SPHERICAL   ASTRONOMY. 

Af  sin  «       .  3-OI4Q4 

.  3-  28473  <•-.. 


T 


2-95424    ^PP-  time  true  d      •     ii  42  56  at  the  place 


8°  4'  .     %   .          2.68445  2  2i  23 

cos      ...  '•'      9- 99568     Equation  ....  3  56 

tan  «J       .      .          9«536i5     App.  time  true  d      .       225  19  at  Greenwich 

tan  I       .     ;j          0.45953  i  time 

"5     ; Long,  in  ] 

I       .     .  N.  70  5i  (  8Pace 

Reduction    .  7 

Latitude.      .  N.  70  58 


Thus  we  find  the  required  place  to  be  in  longitude  E.  1 39°  a4'  and  latitude 
N.  70°  58',  where  simple  contact  will  have  place  at  sunset  and  again  at  sunrise ; 
also  the  middle  of  the  eclipse  would  be  seen  at  midnight  if  it  were  not  intercepted 
by  th%  opacity  of  the  earth.  The  duration  of  the  eclipse  will  correspond  with  the 
duration  of  the  night,  and  therefore  no  portion  of  it  will  be  visible. 


IV.    PLACES   WHERE   THE   MIDDLE   OF  THE   ECLIPSE   HAS    THE    SUN   IN 

THE  HORIZON. 

In  the  present  case  n  is  >  P'  —  (s  +  a)  and  <  *  +  «,    We  must  therefore  pro- 
ceed as  in  No.  19. 

1.  For  the  extreme  points, 

3.52io6 
3.5n84 


°-       .     .     3.  84696  (i) 


+  25  34     ........     3.i8594 


—   55     .......     —2-48430 


.      — 8.97246 
.      .     9.99808  (2) 

*  56  389     .     .     .     3-845o4  (i)  +  (a) 
2     i  27  time  of  middle 

a  —  ir5i2  o448  time  of  beginning 

6  +      5  34  3  58     6  time  of  ending 


APPENDIX    XI. 


367 


PLACE  OF  BEGINNING,  OR  FIRST  EXTREME  PLACE. 

h.    m.    s. 


cos  a  .      . 

cos  S  . 

sin  I  . 

/  .      . 
Reduction 

Latitude 

cos  b  . 
cos  6  .      . 

sin  I  .      . 

I  .      . 
Reduction 

Latitude 

—  9.62918 

tan  a 
sin  i   . 

tan  7* 

h       . 
H     . 

+  0.32738 
+  9.51191 

Greenwich  time 
Equation 

j  time 
(  space 

PLACE. 

Greenwich  time 
Equation     . 

{time 
space 

o    4  48 
3  56 

—  9-60494 

—  o«8i547 

0       ' 

81  18 

o    8  44 

2°   II' 

0        ' 

S.  23  45 
8 

+      2    II 

h.    m.    s. 
3  58     6 
3  56 

S.  23  53 

PLACE 
+  9.39664 

Longitude   W.  83  29 

OF  ENDING,  OR  LAST  EXTREME 

tan  6-     .      +0.58943 
sin  i  .      .      +  9.51191 

tan  h       .     —  1.07762 

+  9.37240 

0 

N.  i3  38 
5 

4       2       2 

60°  3i' 

h 

H      . 

Longitude 

0       ' 

+  94  47 
60  3i 

N.  i3  43 

E.  34  16 

2.  For  the  extreme  times, 


ePf 


the  value  of  rx  taken  out  from  the  preceding  logarithm  of >  is  ih  $7™  10* 

h.    m.  s. 

2  i  27  time  of  middle 
i  57  10     .      .     TJ 

o    417  first  appearance 

3  58  37  last  appearance 


PLACE  OF  FIRST  APPEARANCE. 


sm  i  . 
coat  .     . 

sin  I  .     . 

I  .      . 
Reduction 

Latitude 

+  9«53oro 
+  9.97576 

cot  i    . 
sin  S    . 

tan  h  .      . 

h  .      . 
II       . 

Longitude 

+  0.44339 
+  9.51191 

—  9-5o586 

—  0-93148 

S.  1  8  42 

7 

—  83  19 

+      23 

S.  18  49 

W.85  22 

PLACE  OF  LAST  APPEARANCE. 

Latitude  N. 

i°849 

0       ' 

—   83  19 
1  80    o 

h       . 
H     . 

+    96  4i 
+    60  38 

fa.  m.    8. 

Greenwich  time  o    4  17 
Equation        .  3  56 

(  time      o    8  i3 
Hint 

(  space         2°  3' 


Longitude  K    36    3 


h.  m.    s. 

Greenwich^ time    3  58  37 
Equation  ~.     .          3  56 

{time       4     2  33 
-r-rTcf 
space       oo    38 


368 


SPHERICAL    ASTRONOMY. 


For  the  computation  of  places  in  this  line,  we  have  therefore  the  whole  rang* 
between  the  Greenwich  mean  times  oh  4m  i?8  and  3h  58m  37*.  As  an  example, 
take  the  time  ih  3om. 

h.  m.    B. 

Time  of  middle  2     i  27 
i  3o 


t      . 

.     o  3i  27 

3.27577 

(—0 

c-±!  . 

—  i  o  4o              n 

3-84696 

a 

1  5  34     .        sin 

9.42881 

a     . 
b     . 

.      .  —  35  23 
.      .  —    4  i5 

h.  m.    s. 

cos  a  . 

+  9.91132 

tan  a  . 

.     _9.85r4o 

Greenwich  time 

i  3o     o 

cos  8  .      . 

+  9-97576 

sin  S   . 

.      +  9.51191 

Equation 

3  56 

sin  I    . 

+  9-88708 

tan  h  . 

.      +  0-33949 

{time 

i  33  56 

I 

0 

N  5o  27 

h 

o        / 

n4  35 

space 

23°  29' 

Reduction 

ii 

H 

.      +    23  29 

Latitude 

N.  5o  38 

Longituc 

le     W.  i38     4 

By  similarly  using  the  angle  b,  we  shall  find  the  position  for  the  interval  3im  27' 
after  the  time  of  middle,  or  for  the  time  2h  32m  54s;  thus, 


cos  b  . 
cos  S   . 

sin  /    . 

Reduction 
Latitude 

+ 
+ 

+ 

N. 

9.99880 
9-97576 

tan  6  . 

sin  S   • 

tan  h  . 

h  . 
H 

.     —8 
.     +9 

•     +9 

•87106 
.51191 

.35915 

Greenwich  time 
Equatior 

itime 
space 

2  3  a 
3 

54 
56 

9«  97456 

2  36 

5o 

0        , 

70  35 

7 

o       / 
67     7 
39  i3 

39° 

i3 

N.  70  42               T  ftn_;fn  j-  j  W.  206   20 

The  places  may  be  computed  by  two  together  in  this  way ;  and  it  will  perhaps 
be  a  little  more  convenient  to  assume  a  value  of  t  in  the  first  instance.  We  may 
take  any  value  which  does  not  exceed  rl  or  ih  57m  io8.  In  the  present  example 
we  should  take  t=  3im  27s,  and  begin  as  under: 


•«)• 
«  . 

a  . 

b  . 

0        ' 

.     —  19  49 
i5  34 

log* 
sin  co 

'  cP' 

n 

3.27677 
3-84696 

.     —35  23 
.    —  4  i5 

9.42881 

and  then  proceed  for  the  places  as  above. 


h.  m.    s. 

Time  of  middle    .2127 
t  .      .      .         3i   27 


Time  before  middle  i  3o     o 
Time  after  middle     2  32  54 


APPENDIX    XI. 


369 


V.    NORTHERN    AND    SOUTHERN    LIMITS. 
1.  FOR  THE  PARTIAL  PHASE,  we  have  only  southern  line  of  simple  contact. 

Constants  E,  cos  w,  D',  a'. 
1 4'  56" 


*+6" 


i5  5o 


A'     .    . 

n  . 

n—  A' 

.     3o  46 
+  25  34     ... 

+  3.18594 

—  2.  4941  5 
3-5n84 

—     5    12       ... 

c     . 

E   .     . 
A'        . 

COS  t 

log  w    . 
u    . 
.5     . 
D' 

—  0-69179 
3.52106       P'    .     . 

—  7.  17073        cos  w 

3.26623    .    .    .    . 

+  9.97349         sin  i 

—  8.98231 

3-26623 
+  9«53oio 

+  3.23972 

28'  57"       cos.Z>'    . 
o 
+    1  8   57    59           log  a' 

+   2-79633 
+  9.97448 

—  2-82185 

+    19   26   56                 a'       . 

—     ii'  4" 

The  extreme  places  will  be  the  same  as  those  which  have  the  middle  of  the 
eclipse  with  the  sun  in  the  horizon,  page  366  ;  and  we  mav  compute  for  any  time 
between  the  corresponding  times  of  beginning  and  ending,  viz.  :  oh  4m  48s  and 
3h  58m  63;  or  we  may  take  any  value  of  t  less  than  ih  56"'  39*.  For  an  example, 
take  t  =  o'1  58m  33s. 

3'  54568 
—  7-17073 


Time  of  middle 
t 

Before  middle 
After  middle 


h.  m.    3. 

2  i   27 

0  58  33 

1  2  54 

3  o     o 


—    19  49 
100  53 


E 


M 


—  120  42 
-f     81     4 

+    3o°35'-3 


tan 


COS  U) 
COS  W 

sin  Z 
tan  Z 


Remaining  calculation  for  the  time  3*>  om  o8. 


e+  5  i4-7 
D'  +  19  26-9 
jy  +  T4  4i-6 


— a'    +  32   37-2 
'  —         1 1  •  I 


+  9-77167 
+  9-19113 

+  8-96280 
+  8-96098 
+  9-95835 


32    26-1 


tan  Z     .  . 

cos  M    .  , 

tan  e      .  . 

sin  e       .  . 

cos  .     .  . 

tan  M  .  .      +  Q.8o357 

tan  .     .  .      +  9-80620 

cos  .      .  .      +  9-92544 
tan  (e  +  Dr)    +  9-66258 

tan  / 


sin  Z  . 
cos  M 


—  0-71641 

—  9-27571 
—8-98231 

-f  9-70660 
+  9-77167 

+  9-70660 
+  9.  19113 


Redaction 
Latitude 


+  9.58802 
o      , 

N.  21     10-2 

7-6 
N.  21   18- 


+  8-89773 

Comp.  cos  (h  —  a')  +  0.07456 
Comp.  cos  /  .  .  +  o.o3o35 
check  +  9-00264 


h.    m     & 

Greenwich  time     3     o    o 
.  3  56 


Equation 


time  .  3  56 


24 


space  +  45°  59' 
h     .     .     .     +32    26 
Longitude      .     W.  i3    33 


370 


SPHERICAL   ASTRONOMY. 


The  calculation  for  the  time  ih  am  54*  is  to  be  performed  in  this  manner,  with 
the  same  values  of  tan  Z,  sin  Z,  only  taking  the  value  of  M  =  —  120°  42'. 


A   MORE   ACCURATE   CALCULATION   FOR  THE  TlME   3h   Om   O8. 

Constants  (A\  (£). 

i     .        .        .        .        +   3-220II 


cosD  .     . 

const.  .      . 

9-97456 

A  .    . 

.     +  2.75128 
o«582o4 

3.77671 

3-33332 

(^1)      .     .      .     +  i°39'4o"         (B)  .     . 

These  constants  may  serve  for  the  computations  at 
example  the  following  is  the  process  employed  : 

O          1            II                                                                             t            II 

D     .           4-  19  3r  34  \(j)\  a     •     •     +  ll  49'O 
a  corr.    .                       i  i 
t    ;  i     .      f-  18  58  21            a     .      .      +  3.02898 

(x)  .     .      t-    o  33  i4            cos(D)      +9-97428 

.    +  o°  35'  54" 
all  times.    For  the  present 

P  —  *  ..    .         3-5i255 
p      .     .              9.99982 

log  (*)  - 
ftin  S 

(x)  sin  3 
(A).      . 

P'  cos  5 

X  cos  v  • 

2        •       - 

2  X  COS  V 

•9 
HUg.        . 
«' 
f             •. 
A'. 

f  3-29973 
•L-  9«5i2o4  . 

log  (</)       + 

3-oo326 

P'    .     .               3.5i237 

COS  i        .       .            9.97574 

+  2.81177 

(y)  sin  S    + 
(£)      .     + 

f             + 

2.5i53o 

P'cosJ      .         3.488ii 

+  0°  10'  48" 
f  i     39,40 

o°  5'  28" 
o  35  54 

cos  Z     .      .          9-93493 
2  X  cos  »     .         o-6343o 

+i     5o  28 

o  3o  26 

sin2^     .      .          9«3oo63 
sin  ^      .      .          9-65o3-j 

+  3-82i38 
3-488n  . 

I  log  .     + 

3-26i5o 
3-488ii 

0      ...        26°  33'-  1 
20  ...        53     6-2 

X  sin  v  .      + 
X  cos  v  •      + 

+  0-33327 
o«3oio3 

9.77339 
0-33327 

COS  2  0   .        .              9-77843 

+  o.6343o 

i    a 
.     i4  5o 

T2 

.        .       15       2 

.     .     i5  5o 

tan  v    .      + 
t      .      .      + 

cose'      '.      '.      + 

24°  38'-  9 

3-26764  * 
9.9585i 

(  tant'     .      .      +  9-66169 

J  cos  t      .      .      +  9-9585i 
^ 

(  sin  i'      .      .      +  9-62020 

.     .     3o  52 

sin  t     .     •      -  9-62020 

M        .        .         .        + 
i         .        .        .        + 

3.226i5 

2-88784 
cos  D'                9-  9745  1 

o°28'    3" 
18  58  21 

2.9i333 

IQ   26    24 

APPENDIX    XI. 


371 


D    .     .     .     +  i°9  3  1  34 

(a  —  a')  corr.                           3 

o'    .       .       . 
a     ... 

f-.il/.    . 

(log       . 

.       COS 

y    .     .      . 

j  tan       .     . 

(sin        . 
P'        .      . 

(sin        . 
(cos       .      . 

sin  Z      .      .      . 

-o°r3'3o' 
+  o  17  49 

+      3i  28 

+  3-27600 
+  9.97428 

•  .     .     +    6  35-9 
D'      .     +  19  26-4 
e+  j)t    4.  26     2-3 

O          ' 
h        a    +  36      I  •! 

(D)       . 

£'  .     . 

* 

M    .     . 

Z     .     . 

tan  Z     .     . 
cos  M    . 

tan  0      .      . 
sin  6       .      . 
.  cos  . 

tanJ/   .      . 
{tan  .      .     . 
cos  . 
tan  (0  +  D'} 
tan  I      .      . 

Z       .      .      . 
Reduction   . 
Latitude 

.     -f  19  3i  37 
.     +  19  26  24 

+  3-25028 

+  2.49554 
+  0-75474 

.     -f    o     5  i3 
.     +80°    i'  4 

.     +33°  43  -9 

+  9.82459 
+  0-23864    , 

+  9-9933« 
+  3-5i237 

+  3-5o575 

+  9-74453 
+  9'9'994 
+  9-74453 
+  9.28864 

+  8'-  983  1  7 
')+  .0*092  1  4 
+  o-o3i5i 
+  .9-10682 

h.    m.   s. 
3     o     o 
3  56 

comp.  cos  (h  —  a 
comp.  cos  I 
check     .      . 

Greenwich  time 
Equation    . 
{time   . 
space 
h      ... 
Longitude 

+  9-o6323 
+  9.95352 

+  9-  10682    . 
+  0.75474 
+  9.  86  r  56 

-        i3-7 
h    .    +35  4y4 

+  9.90786 
+  9-68892 

+  9-59678 
N.  21°  33'.  7 

7  -7 

+    3     3  56 

+  45°59'.o 
+  35    47  -4 

N.  21    4i  .4 

W.  10    u  -6 

This  result  differs  materially  from  the  former  one ;  but  we  are  not  to  infer  that 
the  former  position  is  so  far  wide  of  the  truth.  In  general  the  second  determination 
may  be  considered  as  an  almost  accurate  point  in  the  limit,  and  though  the  first 
result  be  some  distance  apart,  yet  it  will  always  be  very  near  to  the  limiting  line, 
sufficiently  near  indeed  for  the  mapping  of  the  lines.  By  direct  calculations  of  the 
eclipse  for  these  places,  the  former  will  have  an  eclipse  of  about  -,,-J^  of  the  sun's  diam- 
eter, and  the  latter  about  y^-J^  of  the  diameter,  which  is  too  small  to  be  perceptible 

2.  FOR  THE  ANNULAR  PHASE,  we  have  both  northern  and  southern  limits. 
Constants  E,  cos  w,  D',  a',  for  northern  limit. 
1 4'  56" 


a  +6" 


A 
n  . 


1 5  5o 

o  54 
+  25  34 
+  26  28 


+  3-18594 

+  3-20085  . 

P'  .  . 
cos  w  . 

+  3-  20086 
+  3-51184 

+  9-98509 
3-52io6 

+  6  464o3 

+  9-68901 

372 


SPHERICAL    ASTRONOMY 


r     ., 
A' 

COS  t 
logM 

u 
S     - 
D'  . 

w 

sin  w 
eP1 

h.    m.    s.              n 

i  42  14    ... 

i.73239     .     .     . 
.    +    9.97349        sin* 

.     +  60°  44-  8 
.     +  9-94075 
.     +  3.846o6 

.      +  3-78771 

1.73239 
+  9«53oio 

.    +     1.70588 
.    +    o°   o'5i"      cos/>' 
.    +  18  57  59 

+   1.26249 
.     +  9.97579 

+   1-28670 

.    +   18  57     8          a'      . 

.     +  o°  o'  19" 

The  semi-duration  of  the  northern  limit  on  the  earth  is  therefore  ih  42m  i4',  and 
we  may  calculate  for  any  value  of  t  not  exceeding  this.  A  calculation  of  the 
extreme  places  on  the  earth  is  to  be  performed  the  same  as  for  the  beginning  and 
ending  of  a  phase  on  the  earth,  and  will  bo  unnecessary  here.  As  an  example,  for 
a  time  within  the  limits,  we  shall  take  t  =  ih  iorn  o8. 


h. 
Time  of  middle   .     2 

t       .      .     i 

m.    s. 
I    27        (- 

IO      O 

«*'   .      . 

«     • 

19  49                E  . 

5o  43           tan  u>'  . 

3-62325 
+  6-464o3 

+  0-08728 

Before  middle     .     o  5i  27    .    .    M  .      .     — 
A  fter  middle        .     3   1  1   27    .     .    M  .      .      + 

Z  .      .     + 
Remaining  cahulation  for  the  time  3h  i  im  27". 

tan  Z     .      .      +  o-o8423 
cos  M    .      .      +  9-93352 

70  32           cos  w'  . 
3o  54          cos  w  . 

,  j  sin  Z    . 
5o  3i-3  ]         7 
(  tan  Z    . 

sinZ.      .      .      . 
cosJ/     .      .      . 

+  9-80147 
+  9-68901 

+  9-88754 
+  0-08423 

+  9-88754 
+  9-93352 

6  +  46 

IO-2 

tan  & 

+  0-01775 

+  9-82106 

J)'  +  18 

57-i 

sin  e 

.      +  9-85817 

comp.  cos  (h  —  a) 

+   0-  l5622 

0  +  D'  +  65 

7-3    . 

cos   . 

+  9-62397 

com  p.  cos  I  . 

+  0-25693 

+  0-23420 

.  check 

+  0-23421 

tan  M  . 

.      +  9.77706 

o 

'            ( 

tan  .      . 

+  0-01126 

A    -  j  '    I   jf  5 

44.6     J 

i. 

a       .      + 

o.3      ( 

cos  . 

.     +  9-84378 

Greenwich  time     3  n  27 

A     .     +  45 

44«o 

tan  (8  +  L 

;')    +  00,3,374 

tan  / 

.     +  0-17752 

{time  . 

3  i5  23 

/ 

.  K56°  2  3'.  8 

space 

+  48°  5i' 

Reduction 

10  «4 

h     ... 

+  45    45 

Latitude 

.  N.  56    34  — 

Longitude 

W.  3     6 

The  calculation  for  oh  5im  27"  is  to  be  performed  in  the  same  manner,  with 
=  —  70°  3a'. 


APPENDIX   XI. 


373 


A   MORE   ACCURATE   CALCULATION   FOR  THE   TlME   3h    II1 
O 


27«. 


D     .     . 

a  corr.    . 
J       .     . 

(*)  •      - 

+  '933  a?l(I))  «     .     . 

+  *8  58  28           a    .     . 
+    o  35     o            cos  (Z>) 

+  a3    6 

+  3-14176 
+  9-97419 

P  —  T     . 

f)      .     . 

3-5ia5S 
9.99901 

log  (x)   . 

sic  3 

+  3-32222              log(y) 
+  9«5i2o8  

+  3.ii595 
+  9.51208 

P'  .    . 

cos  S 

3.5n5ft 
9-97574 

+  2.8343o 

+  2.62803 

FcosS 

3-48730 

(x)  sin  5 
U).      . 

+  0°  n'  23"           (y)  sin  3 
+  i     39  4o             (5)       . 

+  o°  i    5" 
+  o  35  54 

cos  Z     . 

2  X  COS  v 

9-8o33i 

O-6374** 

j 

+  i     5i     3              j 

+  o  28  49 

sin2  0     . 

9-16591 

(log    . 

+  3-82367                 [log. 

+  3-23779 

sin  ^ 

9-58296 

p  cos  i 

3.48730  

3.48730 

(i 

22°  3o'«4 

^  COS  v    • 

+  O.33637               *  sin  "  • 

+  9-75049 

2?     .        . 

45     o  -8 

2         •        • 

o«3oio3              X  cos  v  - 

+  o.33637 

COS  2  0    . 

9.84989 

j   A  COS  V 

+  o  «  63740              tan  v 

4-  o  •  AiAi  a 

4-  o*4{  4i  2 

«'    .      . 

+  20°  9'.  4  . 

/  tan*'     . 
)  cos/     . 

.     +  9-56473 
+  9.97255 

4        .        . 

aug.      . 

<     // 
.     i4  5o 
.     .            9 

{  sin  4'     . 

.     +  9.53-73.8 

«'     . 

• 

.     i4  59 
i5  5o 

A    . 

o  5i      .... 

I  •  7075? 

I  •*7O75'7 

cos  *' 

+  9.97255 

sin  t'     . 

.         +   9.53728 

+  1-68012 

+  1-24485 

w      , 

S      ... 

+   o°   o'48" 
+  18  58  28 

cos  D1 

.     +  9-97577 
+  i  •  26908 

D1    ... 

+  18  57  4o 

a'    .       . 
a     .       . 

.     -f  O°    O'  19" 

.    +  o  a3    6 

D    .     .     . 

+  19  33  27 

(a_a'. 

.     +  0    22    47 

(«  —  o')  corr. 

i 

(log       . 

.     +3-  1  3577 

<D)       .     . 

W  .     .     . 

+  19  33  28 
+  18  57  4o 

COS 

y  •    • 

.     +  9-97419 
.     +  3.10996 

x      .     * 

+    o  35  48    . 

+  3-332<>3 

•*/• 

.     o«o  e_>    T 

!tan 

•      +  9'77793 

M    . 

+  dc    07  «  i  . 

cos 

.      +  9.93328 

P1 

+  3.39875 
.     +3-5u56 

J     .     .     . 

+  5o°  27'.  9 

lain        . 

(  COS 

.     +  9.88719 
.      +  9-8o383 

374: 


SPHERICAL    ASTRONOMY. 


.     +4°6 
.     +  18 

7+65 

'  +  45  4i 
+          o 

t 
5-8 
57-7 
3-5 

•9  - 
•4 

cos  M    .      . 

tan  e       .      . 
sin  0       ^    ff 
.  cos  . 

tan  M    . 
{tan   .      .      . 
cos   . 
tan  (0  +  D'} 
tan  /       .      . 

Reduction  . 
Latitude     . 

+  o-o8336 
+  9-93328 

smZ     .     -*•••. 
cos  M   . 

-1-  9-88719 
+  9-93328 

+  0-01664 
+  9-85764 
+  9-62499 

+  0-23265 
+  9'77793 
+  o.oro58 

+  9-84412 
+  0-33249 

+  9-82047 
comp.  cos  (h  —  a)  +  o-i  5588 
comp.  cos  1       .      +  o-2563o 

.    check    .      .      . 

Greenwich  time 
Equation    . 
{time   . 
space 
h    .      . 
Longitude 

+  o«  23265 
h.    m.  a. 

3  ii  27 
3  56 

+  45  4a 

•  3 

+  o  •  i  766  i 

+    3  i5  23 

K56°  20'-  5 
10  -4 

+  48°  5o'.8 
+  45   42  -3 

N.56   3o  -9 

W.    3      8-5 

VI.    CENTRAL    LINE. 

"We  have,  at  page  363,  found  the  semi-duration  of  the  central  appearance  on  the 
earth  to  be  ih  43m  17*,  which  is  therefore  the  greatest  value  of  t  for  this  phase. 
As  an  example  for  a  time  within  the  limits,  take  the  same  value  of  t  as  in  the  two 
preceding  examples. 

,     3.623*5 
,     3.52io6 


Time  of  middle 

t 

Before  middle 
\fter  middle 


b.  m.    s. 

2       I     27 


o  5i  27 
3  ii  27 


—  19  49 
+  5i  4i 

—  71  3o 
+  3i  62 


Remaining  computation  for  the  time  3h  ura  27s. 


6  +  44  56-o 

i  +  18  58-0 

+  6  +  63  54-o 


+  44  56-6 


tanZ     . 

cos£     . 

tan  e      . 
sin  8 
.    cos  . 

tan  8     . 
j  tan  h     . 

^  cos  h     . 

tan  (e  +  i] 

tan  /      . 

I      . 
Reduction 
Latitude 

+  0-06994 
.      +  9-92905 

.      +  9.99899 
.      +  9.84898 
.      +  9-64339 

+  o«  20559 
.      +  9.79354 

.      +  9-99913 

.      +  9-84991 
).      +  0.30990 

.      +0-15981 

.  K55°  i8'-7 
10  .6 

.  N.  55     20.  - 

t 
,  e 

tan  » 

COS  u, 

ii    . 

A  . 
P"  . 


0-10219 
9-799.46 


sin  Z.      . 

cos  8      . 

comp.  cos  h 
comp.  cos  I 

check 


.  3.39348 

.  3.5ii84 

.  9-88164 

.  0-06994 

+  9-88164 
+  9-92905 

+  9-81069 
+  o-  i  5009 
+  0-24480 


o«2o558 


h.    m.    s. 

Greenwich  time     3  1 1  27 
Equation     .  3  56 

(  time 
space 


h   .      .      . 
Longitude 


+  48°  5r 
+  44  57 
W.T  54 


APPENDIX  XL 


375 


A   MORE   ACCURATE   CALCULATION. 


D    . 
a  corr 

<*)     - 


4- 

IO  33  27  )  /  Tvt  « 

,     .      +  23    6    . 

+  3.14176 

*y  A3  *j  f  /  m 
1  4 

cos  2)    .     .     . 

.      + 

18  58  28 

/        V 

+  3.ii595 

+ 

(x)  . 

+  3.32222 

3-51255               8 

.     +3i°5a'-7| 

;  tan  S    .      .      . 

!  cos  S     .      .     . 

+  9.79373 
+  9-  92899 

•      • 

9.99903 

A    .... 

+  3.39323 

3.5n58      .     .     . 

3-5n58 

Z 

tanZ     . 

.      +  49°  35'  -6- 
.     +  0-06994 

[  sin  Z     .     .      . 
[  tan  Z    .      .     . 

smZ     .      .      . 

+  9.88165 
+  0-06994 
+  9.88165 

o 

cos  S 

.     +  9.92899 

cos  S     .     •     . 

+  9'92899 

+  44 

55-8        tan  0 

.     +  9-99893 

+  9.  81064 

+  18 

58-5         sin  0      . 

.     +  9.84895 

com  p.  cos  h 

+   O«  I5O20 

+  63 

54-3    .    cos    .      . 

.     +  9-6433i 

com  p.  cos  I 

+  0.24480 

+  o-2o564    . 

check    .      .      . 

+  o-2o564 

tan  S    . 

.     +  9-79373 

o 
4-  44 

57.5     I""1*     ' 

-      +  9-99937 

(  cosA     . 

.      +  9.84980 

Greenwich  time 

3  ii  27 

tan  (0  +  3 

)       +  o-3iooo 

Equation     . 

3  56 

tan/      . 

+  0-15980 

(  time     . 

TT  •          ] 

+    3  i5  23 

Z       .      . 

.  N.  55°i8'.7 

(  space    . 

+  48°5o'.8 

Reduction 

10  «6 

h      .     .     .     . 

+  44   57.5 

Latitude 

.  N.  55   29  .3 

Longitude  .     . 

W.   3  53.^ 

CENTRAL  ECLIPSE  AT  NOON. 


Diff.  dec. 
P     . 
smZ      . 

Z      . 
t       .      . 
/       .      . 
Reduction 
Latitude 


.     3.21245 
.     3.5n84 
.     9.70061 

Time  of  6      •      • 
Equation        .      . 
|  time  . 

Long,  in  •{ 
(  space 

h.  m.    8. 
2  ai   23 
+       3  56 

2  a5  19) 
(.  w 

+  3o°    8' 
+    18     58 

36°  20'  ( 

N.  49      6 
ii 

N.  49     17 


By  assuming  a  series  of  times,  and  so  computing,  in  conformity  with  the  preced 
ing  examples,  a  series  of  points  on  each  of  the  several  limits  will  be  determined; 
and  these  points  being  laid  down  in  a  geographical  map,  with  respect  to  latitude 
and  longitude,  it  will  be  easy  to  trace  the  lines  through  them.  In  this  manner  has 
the  following  map  been  executed,  the  assumed  law  of  projection  being  that  the 
parallels  of  latitude  are  concentric  and  equidistant  circles.  This  projection  will 
be  found  ver^  suitable  when  an  eclipse,  as  in  the  present  instance,  extends  com- 
pletely round  one  of  the  poles  of  the  earth.  In  other  cases,  any  hypothesis  what- 
ever may  be  assumed,  with  respect  to  the  law  of  projection,  provided  the  geo- 
graphical sketching  and  eclipse-lines  be  both  laid  down  on  the  same  principle, 
(See  Fig.  11.) 


PRINCIPAL  LINKS  FOR  THE  SOLAR  ECLIPSE  OF  MAY  14-15,  1836 

lass     sg'  *'  s       s        s 


APPENDIX    XL  377 

PHENOMENA    FOR    A    PARTICULAR   PLACE. 
I. — ECLIPSES  OF  THE  SUN. 

The  chief  objects  of  determination  for  any  particular  place  are — 

1.  For  a  partial  eclipse,  its  magnitude,  and  the  times  of  beginning,  greatest 
phase,  and  ending. 

2.  For  a  total  eclipse,  the  times  of  external  and  internal  contact  of  limbs,  or  the 
times  of  partial  and  total  beginning  and  ending. 

3.  For  an  annular  eclipse,  the  times  of  exterior  and  interior  contact  of  limbs,  or 
the  times  of  partial  and  annular  beginning  and  ending. 

Also,  to  secure  certainty  in  the  observation,  it  is  necessary  to  determine,  in  each 
case,  the  particular  points  on  the  limb  of  the  sun,  as  related  either  to  the  vertical 
or  a  circle  of  declination,  where  these  contacts  take  place  ;  and  hence  the  general 
configuration  of  the  ellipse. 

We  first  proceed  to  find  expressions  for  calculating,  at  any  time,  the  apparent 
relative  position  of  the  two  bodies,  and  the  augmentation  of  the  semi-diameter  of 
the  moon.  The  parallax  in  altitude  depends  on  the  Eq.  (8)  or  (9),  page  336.  It 
will  here  be  necessary  to  investigate  the  effects  which  this  parallax  will  produce 
in  the  right  ascension  and  declination  of  the  moon.  These  might  be  accurately 
determined  by  the  theory  of  the  small  variations  of  spherical  triangles,  but  not 
quite  so  simply  as  in  the  following  manner: — Assume,  as  before, 

I,  the  geocentric  latitude  of  the  place  ; 
R.  A.,  the  true  right  ascension  of  the  moon; 

D,  the  true  declination  of  the  moon,  +  north,  —  south; 
h,  the  true  hour  angle  of  the  moon,  +  west,  —  east ; 
r,  the  distance  of  the  centres  of  the  earth  and  moon. 

Then  if,  from  the  earth's  centre,  we  take 

a,  on  the  intersection  of  the  planes  of  the  meridian  and  equator,  -t    towards 

upper  meridian  ; 

y,  in  the  plane  of  the  equator,  +  west,  —  east ; 
z,  parallel  to  the  earth's  axis,  +  north,  —  south  ; 

we  shall  have,  for  the  position  of  the  moon, 

x  =  r  cos  D  cos  h,  y  —  r  cos  D  sin  ht  z  =  r  sin  D ; 

and,  for  the  position  of  the  observer, 

(a;)  =  f>  cos  I,  (y)  =  0,  (z)  =  p  sin  I. 

Thus  the  position  of  the  moon,  in  relation  to  the  observer  as  an  origin,  will  be 

«'  =  x  —  (a;)  =  r  cos  D  cos  h  —  p  cos  I ; 
y'  =  y  —  (y)  =  r  cos  D  sin  h ; 
z'  =  z  —  (z)  =  r  sin  D  —  p  sin  / ; 

and  hence,  D\  h!  denoting  the  apparent  declination  and  hour  angle,  and  ff  Ihtf 
distance  of  the  moon  from  the  observer,  we  shall  have 

x  =  r'  cos  D'  cos  h'  =  r  cos  D  cos  h  —  p  cos  / ; 
y'  =  r'  cos  D'  sin  h'  =  r  cos  D  sin  h  ; 
»'  s=  r'  ain  D'  =  r  sin  D  —  p  sin  I. 


tt) 


378  SPHERICAL    ASTRONOMY 

Therefore,  as  cot  A'  =  -  ,  tan  J}'  =  -  sin  h',  -  =  sin  P,  we  find 

y  y  r 

p  sin  P  cos  I 

cot  A'  =  cot  A =—r — r- 

cos  D  sin  A 

/         p  sin  P  sin  /\  sin  h1 

tan  Z>'  =  I  1  —  " : — =. —  I  -. — r  tan  J) 

\  sin  D      /  sin  A 

or 

/    p  sin  P    \ 

cot  A  —  cot  A'  =  I 77—: — :  I  cos  I 

\co8  D  sin  A/ 

tan  D  _  tanZ)'  __  /     p  sin  P    V       ^ 
sin  A         sin  A'         \cos  .Z)  sin  A/ 

which  present  a  direct  method  of  calculating  the  apparent  position  of  the  moon, 
at  any  time,  from  that  of  the  true.  The  former  of  these  equations  is  evidently 
subservient  to  the  other,  and  must  necessarily  be  computed  first.  As  the  calcula- 
tion of  these  expressions  will,  in  general,  require  seven  places  of  figures,  it  will  be 
more  convenient  to  determine  the  simple  effects  of  the  parallax,  or  the  small  dif- 
ferences A.R.  —  A.R.'t  D  —  D',  for  which  other  expressions  may  be  derived  from 
them.  Let  A.R.  —  A.R.'  =  A'  —  A  =  A  A,  and  D  —  D'  =  A  D ;  then  by  multi 
plying  the  equation 

.  . .       p  sin  P  cos  / 

COt  A  COt  A    =  ;— r- 

cos  D  sin  A 
by  sin  A  sin  A',  the  left-hand  member  will  become  sin  (A'  —  A)  or  sin  A  A. 

p  sin  P  cos  /   . 

.  • .  sin  A  A  = ^r —  sin  A . 

cos  D 

Again  we  have 

tan  D       tan  D1 p  sin  P  sin  / 

sin  A         sin  A'         cos  D  sin  A ' 
But 

tan  D       tan  D'       tan  D  —  tan  D' 


sin  A         sin  A'  sin  A  \sin  A       siu  A' 

_      sin  (D  —  D')   _       sin  h'  —  sin  A  ^ 


sin  A  cos  J9  cos  D'          sin  A  sin  A' 

sin  A  7>  2  sin  £  A  A  cos  (A  +  i  A  A)  4       _ . 

^    H * ; — r— :-; tan  if  . 


sin  A  cos  D  cos  D'  sin  A  sin  A 

«       A     ,  ,    p  sin  P  sin  Z 

Equate  this  with  -  --  77-7—  r,  and  we  find 
cos  D  sm  A 

sin  A  -D      _  p  sin  P  sin  /       2  sin  •£  A  A  cos  (A  +  |  A  A)     sin 


cos  Z>  cos  D'  cos  Z>  sin  A 

sin  A  A         p  sin  P  cos  I        sin  A' 


But  2sm  i  /.  A  = 


APPENDIX    XI.  379 

Substitute  this  value  and  multiply  by  cos  D  cos  D'  and  we  deduce 

sin  A  D  =  p  sin  P  I  sin  /  cos  D'  —  cos  /  sin  D' I. 

L  cos  i  A  A      J 

We  shall  therefore  have,  for  the  parallax  of  the  hour  angle,  and  that  of  the  decli- 
nation, 

(p  cos  Z)  sin  P    .  'J 

sin  A  h  =  — — - sin  h' 

cos  D 

I"  (2) 
sin  A  D  =  sin  P  I  (p  sin  I)  cos  D'  —  (p  cos  f)  sin  D'  -     ^ — —  I 

These  are  still  however  not  adapted  for  direct  calculation,  since  they  involve 
the  apparent  quantities  A',  D',  which  it  is  our  object  to  determine.  The  only  use 
that  can  be  made  of  them  is,  first  to  use  the  true  quantities,  in  order  to  get  the 
parallaxes  and  apparent  values  approximately,  and  then  to  repeat  the  operation. 
To  avoid  this  difficulty,  substitute  in  the  former  A  +  A  A  instead  of  A',  and  in  the 
latter  put  J)  —  AD  instead  of  J)',  and  we  get,  by  expansion, 

p  cos  /  sin  P  .  . 
sin  A  A  =  • — —  (sm  A  cos  A  A  -f-  cos  A  sin  A  A) ; 

sin  A  D  =p  sin  P  cos  AD  I  sin  /  cos  D  —  cos  I  sin  D — '  I 

L  cos  i  A  A      J 

•4-  p  sin  P  sin  A  D  \&a\  I  sin  D  -f  cos  I  cos  D  -— 1— t-"L__J  I 
L  cos  J,  A  A      J 

Divide  these  by  cos  A  A,  cos  A  D,  respectively,  and  solve  for  tan  A  A  and  tan  A  1), 
and  we  find 

/p  cos  /  sin  P\    , 

I  -     -—  1  sin  A 

V       cos  D      / 

tan  A  A  = .     — (8) 

/p  cos  I  sin  P\ 

1-1-        — j, 1  cos  A 

V       cos  D      / 


tan  A  D 


P  sin  P  Fein  /  cos  D  -  cos  /  sin  D  cos  (h  +  JL 
L  cos  \  A 

1  —  p  sin  P  I  sin  I  sin  D  -f-  cos  /  cos  D 


cos  i  A  A 

^  T         tanD    cos  (A  +  |  A  A)~l   '   " 

(p  sm  /  sin  P)  cos  D  I  1 .  J — L- I 

L          tan  /          cos  -J  A  A      J 


/       •      7    •       nx     •       r^r,  1  COS  (A  4-  i   A  A)"l 

1  -  (pain  /sm  P)  sin  D\  1  + . S — 

L       tan  /  tan  D         cos  i  A  A      J 


These  expressions  are  all  of  them  perfectly  rigorous,  and  better  suited  to  calcu- 
lation than  they  would  appear  at  first  sight.  The  process  of  the  calculation,  in 
which. five  places  of  figures  will  be  sufficient,  is  more  detailed  in  the  following 
equations: 

(p  cos  1)  sin  P  n  sin  A 

n  =  —    — L- ;  tan  A  A  = .    .       (5) 

cos  D  1  —  n  cos  A  v  ' 


380 


SPHERICAL    ASTRONOMY. 


c  =  (p  sin  /)  sin  P ; 
nx  =  k  tan  D  ; 


eou 


tan  AD  = 


na  = 
c  cos  D  (1  — 


k 
tan  D 


1  —  c  sin  D  (1  -f  na) 

The  expression  (4)  for  tan  A  -D  may,  however,. be  neatly  resolved  by  means 
spherical  triangle  as  follows: 

Assume  Fig.  9. 


of 


(A)  being  very  nearly  equal  to  h  -{-  £  A  #.  And  let  N  be  the 
north  pole,  Z  the  central  zenith,  and  AT  the  moon;  then  NM 
=  90°  —  D,  NZ  =  90°  —  J,  and  the  /  N  =  h.  "Without 
changing  these  values  of  NM,  NZ,  let  us  suppose  the  hour 
angle  JVto  become  increased  to  the  value  of  (A);  and  with  the 
triangle  so  constituted  suppose  the  altitude  of  the  moon  to  be 
«,  so  that  ZM=  90°  —  «;  then  the  spherical  relations 


will  give 


sin  Z  M  cos  M  =  cos  NZ  sin  N  M  —  sin  NZ  cos  N  M  cos  N, 
=  cos  NZ  cos  NM  +  sin  NZ  sin  NM  cos  N, 


cos  «  cos  M  =  sin  J  cos  D  —  cos  ^  sin  D  cos 


•     i         T^  i   •     T-K 

=  sin  I  cos  JD  —  cos  /  sin  .Z) 


4-  •$•  A 


cos   A 
sin  e  =  sin  /  sin  D  -\-  cos  /  cos  D  cos  (^) 

.  ,  .  r,  ,        r,  cos  (A  +  -i  A  A) 
=  sin  /  sin  D  +  cos  I  cos  .D  --  ^—  ——  -  :. 

cos  -J  A  h 

Comparing  these  with  the  former  expression  of  (4),  we  have  therefore 


tan  A  D 


(p  sin  JP)  cos  e 


.  cos  M 


1  —  (p  sin  P)  sin  t 

Before  this  can  be  used  the  angles  M  and  e  must  be  determined. 
Draw  ZD  perpendicular  to  MN,  and  by  spherics, 

tan  ND  =  tsn\NZcos  N    .     .    .    .    ^^    , 
sin  M  D  tan  M  =  tan  ZD  =  sin  N  J)  tan  N; 


Also  by  (c) 


sn 

.  tan  M  =  -  —  —  -  tan  N 
sin  M  D 


ta.nMZ  =  ,  or  cot  MZ  =  cot  M  D  cos  M 

cos  M 


tan  M       cos  N    sin  M 


sin  MD  ~~  tan 


cof'  sin  7 


cos  N  sin  NZ 
co»  jTsin  MZ 


(d) 


APPENDIX    XI.  381 

Let  now  ND  =  0,  and  M D  =  MN—  0  =  90°  — -  (0  -f  D) ;  and  the  equations 
(a),  (6),  (c),  (d),  (4  (/),  will  give  the  following: 


__  cos  (A  -f  \  A  A) 

(a\ 

tan  0 

cos  -£  A  A 
~~~  cot  1  cos  (A)              .     .     .     r 

.  (b) 

8in  *        tan  (A) 

(e\ 

tan  e 

cos  (0  +  D)  tan  *  ' 
—  tan  (0  -f-  D)  cos  3f  .     .     .    . 

.  (d) 

sin  0 

cos  (h)  cos  I 

(e\ 

cos  (0  -f-  D) 
tan  A  D 

cos  Jlf  cos  « 
(p  sin  P)  cos  e 

(  f} 

1  —  (p  sin  P)  sin  e 

•     •  V/  ) 

in  which  the  equation  (e)  is  used  as  a  check  on  the  preceding  computations.  This 
check  affords  a  good  security  to  the  accuracy  of  the  work,  and  gives  to  these  equa- 
tions a  decided  preference  over  those  of  (6),  although  a  trifle  more  perhaps  in  point 
of  calculation.  They  have  also  another  advantage,  inasmuch  as  M  may  be  consid- 
ered as  the  parallactic  angle,  and  c  the  altitude  of  the  moon ;  the  former  of  these 
is  useful  in  determining  the  position  of  the  line  joining  the  centres  of  the  two  bod- 
ies in  relation  to  the  vertical,  and  the  other  is  useful  in  finding  the  augmentation 
of  the  moon's  semi-diameter,  which  we  shall  now  consider. 

If  s'  denote  the  moon's  apparent  semi-diameter,  and  s  her  true  semi-diameter  as 
seen  from  the  centre  of  the  earth,  the  actual  semi-diameter  of  the  moon  will  be 
represented  by  both  r  sin  s,  and  r  sin  s' ;  also,  if  a  perpendicular  be  drawn  from 
the  centre  of  the  moon  upon  the  radius  p  produced,  this  perpendicular  will  be  rep- 
resented by  both  r  sin  Z,  and  r  sin  Z '.  We  must  therefore  have = . 

sin  s        sin  Z 

Let  M  be  the  true  position  of  the  moon,  in  the  preceding  figure,  and  sin  ZM 
sin  /.  NZM—  sin  NM sin  N  will  be  sin  Z  sin/  NZM=cos  D  sin  A;  for  the 
apparent  position  of  the  moon  the  angle  N  Z  M  will  remain  the  same,  and  sin  22 
sin  Z  NZM=co3  D'  sin  A'. 

sin  Z'  .__  cos  D'    sin  A' 
' '  sin  Z        cos  D  '  sin  A  ' 

Also,  by  means  of  the  equations  (8)  and  (9),  page  336, 

sin  Z' p  sin  P  sin  Z'  _  sin  z  cos  z  _  cos  z 

sin  Z       f  sin  P  sin  Z       p  sin  P  sin  Z  ~"  p  sin  P  sin  Z  "~  1  —  p  sin  P  cos  Z 

m  sin  s' sin  Z' cos  J)'    sin  A' cos  z 

"sin  x        sin  Z        cos  D  '  sin  A        1  —  p  sin  P  cos  Z 

All  the  preceding  formulae  are  strict  in  theory.  It  now  remains  to  consider 
what  allowances  may  be  made  and  what  facilities  given  in  their  actual  calculation. 
In  the  first  place  the  value  of  cos  £  A  A  may  be  safely  assumed  equal  to  unity, 
and  may  therefore  be  rejected  in  the  equations  (2),  (4),  (6),  and  (7),  so  that  (A)  = 
A  +  i  A  A ;  it  may  be  shown  that  this  supposition  cannot  involve  an  error  of 
more  than  0".03  in  the  value  of  A  D. 


SPHERICAL    ASTRONOMY 
Also,  as  the  arcs  P,  A  A,  A  D,  are  small,  we  must  have  very  nearly 
^  =  sin  1"  =  [4.68657],     *!LA*  =  ta^J  =  tan  1"  =  [4.68657], 

where  P,  A  h,  A  D,  denote  respectively  the  numbers  of  seconds  they  contain. 
These  equations  may  be  made  more  exact,  for  the  limits  between  which  the  angles 
are  always  comprised,  by  adopting  numbers  differing  a  little  from  sin  1"  and 
tan  1";  thus,  by  assuming 

^  =  [4.68655],  '-^  =  [4.68561]. 

Jr  A  A 

The  first  supposition  will  not  in  any  case  involve  an  error  exceeding  that  of 
0''.05  in  the  value  of  P,  nor  the  second  an  error  of  more  than  0".l  in  the  value  of 
A  A,  and  these  are  much  too  small  to  merit  attention ;  the  latter  assumption  ap- 
plies equally  the  same  to  A  D. 

Thus  we  shall  have  (A)  =  h  +  i  A  h,  sin  P=  [4.68555]  P,  A  A  =  [5.31439] 
tan  A  A,  A  D  =  [5.31439]  tan  A  D ;  also,  A  h  =  A  a,  the  parallax  in  right  as 
cension.  The  equations  (3)  and  (7)  may  therefore  be  commodiously  arranged  as 
follows : 

c  =  [4.68555]  p  ; 
A  =  c  P ;  m  =  A  cos  /  ;  .   k  = 


cos  D  f  .     .     .  (9) 
n  =  kcosh;  A  «  =  [5.31439]  ^^J 

By  taking  h  less  than  180°,  positively  or  negatively,  A  a  will  have  the  same 
•ign  as  h. 


tan  0  =  cos  (A)  cot  I ;  G  =  cos  (A)  cos  I 

tan  M  = ,„  ,    y..  tan  (A) ;  tan  c  =  tan  (0  -f  D)  cos  M  \ 

^   .  •  (10) 

sin  0  G 

5  =  cos  J/cos  e.  check  .     .     .  — — , — —  =  - 

cos  (0  +  jD)        £ 

A    /? 

wi  =  ^1  sin  t ;  A  7)  =  [5.31489] 

1  —  Wi 

The  auxiliary  arc  0  may  be  taken  out  in  the  first  quadrant,  -f-  or  — ;  calling  0°  to 
180°  the  first  semicircle,  and  180°  to  360°  or  0°  to  —  180°  the  second  semicircle, 
the  parallactic  angle  M  must  be  taken  out  in  the  same  semicircle  with  A ;  and 
A  D  will  have  the  same  sign  as  cos  M. 

It  will  appear  by  the  preceding  investigations  that  the  values  of  A  a,  A  D,  so 
deduced,  are  the  quantities  to  be  subtracted  from  the  true  values  of  A.R.,  D,  to 
get  the  apparent. 

As  the  number  n  is  always  very  small,  the  values  of  comp.  log.  (1  —  w)  to  the 
fifth  place  of  figures  may  be  comprised  in  the  following  useful  Table  under  the 
title  of  Correction  of  Log.  Parallax,  and  conveniently  taken  out  with  the  nearest 
third  fig-ire  of  the  argument. 


APPENDIX   XI. 


383 


Correction  of  Log.  Parallax. 

Argument:  log.  n. 

Log  n 

Corr. 

Log  n 

Corr. 

Log  n 

Corr. 

Log  n 

Corr. 

1 
Logr* 

Corr. 

5-00 

o 

7'ioo 

54 

7.400 

109 

7-700 

218 

8-000 

436 

•  10 

0 

•  no 

55 

«4io 

112 

.710 

223 

•010 

447 

•20 

I 

•  i  20 

57 

•420 

u4 

•  720 

229 

«O20        457 

.3o 

I 

•  i3o 

58 

•  43o 

117 

•  73o 

234 

•  o3o 

468 

.40 

I 

*i4o 

60 

•44o 

1  20 

.74o 

240 

«o4o 

479 

•  5o 

I 

•  i5x> 

61 

•45o 

123 

•  75o 

245 

•o5o 

490 

.60 

2 

.160 

63 

•46o 

125 

•  760 

25l 

•060 

5oi 

.70 

2 

•  170 

64 

.470 

128 

.770 

257 

•070 

5i3 

•  80 

2 

«i8o 

66 

•48o 

i3i 

.780 

263 

•080 

525 

.90 

3 

•  190 

68 

•  490 

1  34 

.790 

269 

•090 

537 

6-00 

4 

•  200 

69 

•5oo 

i37 

•  800 

275 

«IOO 

55o 

•  10 

6 

•210 

7i 

•5io 

i4i 

•  810 

281. 

•110 

563 

•  20 

7 

•  22O 

72 

•520 

1  44 

•  820 

288 

•I  20 

576 

•  3o 

9 

.230 

74 

•53o 

1  48 

•  83o 

294 

•  i3o 

5oo 

•  4o 

ii 

•24O 

76 

•54o 

i5i 

•  84o 

3O2 

«i4o 

6o4 

.5o 

i4 

•  25o 

77 

•55o 

i55 

•  85o 

3o8 

•  i5o 

618 

.60 

17 

.260 

79 

•  56o 

1  58 

.860 

3i5 

.160 

632 

•  70 

22 

.270 

8k 

•57o 

162 

.870 

323 

•  170 

647 

•  80 

27 

.280 

83 

•  58o 

1  65 

•  880 

33i 

•  180 

663 

.90 

34 

.290 

85 

•59o 

169 

•  890 

338 

-190 

678 

7.00 

43 

.3oo 

87 

•600 

i73 

•  900 

346 

•200 

694 

7-000 

43 

•  3io 

89 

•  610 

177 

•910 

355 

•2IO 

710 

•010 

44 

•32O 

91 

•  620 

181 

•  920 

363 

•  22O 

727 

•  020 

46 

•  33o 

93 

•  63o 

1  86 

.93c 

37i 

•230 

744 

•  o3o 

47 

•  34o 

95 

•  64o 

191 

.940 

379 

•  240 

761 

•  o4o 

48 

•  35o 

98 

•  65o 

i95 

•  95o 

388 

•250 

779 

•  o5o 

49 

•  36o 

IOO 

•  660 

199 

.960 

398 

8-260 

79& 

•  060 

5o 

•  37o 

IO2 

•  670 

204 

.970 

407 

•  070 

5  1 

•  38o 

io4 

•  680 

209 

•  980 

4i7 

•  080 

52 

•  39o 

107 

•  690 

213 

7.990 

427 

•  090 

53 

7.400 

109 

7.700 

218 

8-000 

436 

7.100 

54 

This  correction  is  additive  when  n  is  positive,  and  subtractive  when  n  is 

negative.     For  the  parallax  in  declination  it  will  always  be  additive  if  the 

moon  be  above  the  horizon. 

For  the  augmentation  of  the  moon's  semi-diameter  we  may  assume  cos  z  =  1  and 
Z  =  90°  —  c,  so  that 


«       1  —  p  sin  P  sin  «       1  —  ni ' 

ni  being  the  number  which  enters  into  the  computation  of  A  D.    Hence 
,  _      s       __  [9.43537]  P 


(H) 


384:  SPHERICAL    ASTRONOMY. 

This  and  the  last  formulae  for  A  a,  A  -Z>,  entirely  preclude  the  necessity  of  having 
recourse  to  a  table  of  the  sines  and  tangents  of  small  arcs,  and  possess  much  uni- 
formity and  simplicity  in  their  application. 

To  get  the  relative  parallax  of  the  moon  with  respect  to  the  sun,  we  must  use 
P  —  IT,  instead  of  P.  If,  therefore,  P'  denote  the  value  of  p  (P  —  TT),  or  the  rela- 
tive horizontal  parallax  reduced  to  the  latitude  of  the  place,  we  must  use  sin  P', 
instead  of  p  sin  P,  in  the  preceding  formulae. 

The  determination  of  the  apparent  relative  positions  of  the  centres  of  the  two 
bodies,  as  well  as  the  augmentation  of  the  semi-diameter  of  the  moon,  at  any  time, 
has  now  been  reduced  to  a  practical  and  expeditious  set  of  formulae.  A  series  of 
these  apparent  positions  of  the  moon,  with  respect  to  that  of  the  sun,  will  trace 
out  her  apparent  relative  orbit;  and  the  contact  of  limbs  will  evidently  take  place 
when  the  apparent  distance  of  the  centres  becomes  equal  to  the  sum  or  difference 
of  the  semi-diameter  of  the  sun  and  the  augmented  semi-diameter  of  the  moon. 
For  a  distance  equal  to  the  sum  of  these  semi-diameters  we  shall  have  partial  be- 
ginning or  ending;  for  a  distance  equal  to  their  difference  we  shall  have 

Insular  \  beginning  or  endinS>  when  '{<:[* 

Since  the  hour  angle  of  the  bodies  is  subject  to  the  rapid  variation  of  nearly  15° 
per  hour,  the  effect  produced  by  parallax  will  be  of  so  irregular  a  nature  as  to 
give  a  decided  curvature  to  the  apparent  relative  orbit  of  the  moon.  This  curva- 
ture will  be  more  strongly  characterized  when  the  eclipse  takes  place  at  some 
distance  from  the  meridian  or  near  to  the  horizon ;  and  the  apparent  relative 
hourly  motion  of  the  moon,  even  during  the  short  interval  of  the  duration  of  the 
eclipse,  will,  through  the  same  irregular  influence,  experience  considerable  varia- 
tion. These  circumstances  will,  in  some  measure,  vitiate  any  results  deduced  in 
the  usual  manner,  by  supposing  the  portion  of  the  orbit  described  during  the 
eclipse  to  be  a  straight  line,  and  using  the  relative  motion  at  the  time  of  apparent 
conjunction  as  a  uniform  quantity.  The  method  we  are  about  to  pursue  is  very 
simple,  and  consists  in  assuming  any  time  within  the  eclipse,  and  computing  for 
this  time  the  relative  positions  and  motion  of  the  bodies,  and  thence  finding,  with- 
out any  reference  whatever,  either  to  the  time  of  the  middle  of  the  eclipse  or  to 
the  time  of  conjunction,  the  times  of  beginning,  greatest  phase,  and  ending,  and 
the  relative  positions  of  the  bodies  at  these  times.  The  nearer  the  assumed  time 
is  to  the  time  of  the  greatest  phase,  the  more  accurately  will  the  time  of  that 
phase  be  determined  ;  and,  similarly,  the  nearer  that  time  is  to  the  time  of  begin- 
ning or  ending,  the  more  certainty  will  attach  to  the  determination. 

To  find  the  apparent  relative  motion  of  the  moon,  we  must  first  determine  the 
variation  which  takes  place  in  the  parallax.  For  this,  take  the  equations  (2),  p, 
879,  viz.: 

.       sin  P'  cos  I  .      ,  * 

sin  A  a  =  sin  A  h  = — —  sin  h, 

cos  D 

sin  A  D  =  sin  P'  Fsin  /  cos  D'  -  cos  /  sin  D'  °™  (*  +  *  ^  *)"| . 
L  cos  i  A  h      J 

or,  substituting  small  arcs  instead  of  their  sines, 


A  D  =  P'  [sin  /  cos  D'  -  cos  I  sin  V  ™  (k  +  i  **>]. 

cos  i  A  h      J 


APPENDIX   XI.  385 

Since  a  portion  of  the  apparent  disk  of  the  moon  is  projected  on  that  of  the  sun, 
the  apparent  declination  D'  can  differ  very  little  from  &.  As  the  hourly  variations 
of  these  small  quantities  are  only  required  approximately,  we  may  therefore  use 
o  instead  of  D1  and  neglect  A  A,  so  as  to  have 

r,,  cos  I    . 

A  a  =  P  --  -  sm  h, 
cos  D 

A  D  =  P'  (sin  I  cos  S  —  cos  I  sin  <3  cos  A)  ; 

which  vixpressions,  though  rough  values  of  A  a,  A  D,  will  give  their  hourly  varia- 
tions pretty  accurately.  For  these,  observing  that  h  is  the  only  quantity  which, 
by  its  rapid  variation,  has  any  sensible  influence  on  these  values,  we  have  by 
differentiation, 


dt 


a)       /      dh   .       \  cos/ 
'  =  I  P1  —  sin  1"  1  -  -  cos  h, 
\      dt  /  cos  D 

/r>,dh    .      ,,\ 
=  IP'  -j-  sm  l'1)  cos  /  sm  t  sin  h. 


Bat  by  the  equations  (9), 

m  =  [4.  68555]  P'  cos/, 

«  =  [4.68*55]  P'  -^-£  cos  A. 
cos  2> 

Suetitute,  therefore, 

P'^LL  cos  *  =  [5.31445]  «, 
cos  D 

P'  cos  /  =  [5.31  445]  m; 


^A-}=  [5.31445]  (^  sin  1")  m  sin  I  .in  ». 


If  we  adopt  14°  29'  as  a  mean  value  of  —  ,  we  shall  have  —  sin  1"=  [9.40274] 
and  [5.31445]  (^  sin  1")  =  [4.71719]  or  [4.7172].     Therefore,  if  (<J),  the  value 


of  the  sun's  declination  at  the  time  of  the  middle  of  the  eclipse,  be  adopted  in  tfie 
value  of  -i-j  -  -,  we  may  form  the  constants, 

Oi  =  [4.7172],  ) 

&  =  [4.7172]  m  sin  (<5)f    ' 


and  then,  using  A  a,,  A  A  in  place  of  -  —    —  \  -ve  shall  hare 

at  dt 


which  offer  a  simple  calculation. 
25 


386  SPHERICAL   ASTRONOMY. 

Let  now,  at  any  assumed  time  within  the  Fig.  10. 

duration  of  the  eclipse,  S  and  M  be  the  ap- 
parent positions  of  the  centres  of  the  sun 
and  moon  ;  and  B  M  E  an  arc  of  a  great  cir- 
cle coinciding  with  the  relative  direction  of 
the  moon's  motion  at  that  time,  which  arc 
we  shall  first  adopt  in  place  of  the  curvilin- 
ear orbit  actually  described.  On  the  circle 
of  declination  S  JV,  demit  the  great  circle 

perpendicular  M  d,  and  suppose  B  and  ^to  be  the  positions  of  the  moon  at  the 
respective  times  of  partial  beginning  and  ending  of  the  eclipse,  and  n  the  middle 
point.  Assume  SB=8E=  s'  +  a  —  A',  8d  =  x,  d  M  =  y,  SM  —  W  Sn  =  n, 
Z.NSM  =  S,  /.BMd  —  /_  dSn  =  i,  and  the  /.£Sn  =  /.ESn  =  *.  Also, 
for  simplicity,  let  x\,  y\  denote  the  hourly  variations  of  x  and  y. 

In  determining  the  value  of  x  we  shall  require  the  a  correction,  which  will 
reduce  the  declination  of  the  point  M  to  that  of  d.  This  correction  is  shown  in  a 
table  at  p.  342  ;  but,  as  this  small  correction  may  be  wanted  more  accurately 
than  can  be  obtained  from  that  table,  we  shall  here  give  some  factors  for  its  de- 
termination, from  which,  in  fact,  the  table  alluded  to  has  been  derived  The  cor- 
rection will  resolve  as  follows: 


COS 

tan  D 


cos  a 


cos  a 
Or,  supposing  cos  a  =  1  in  the  denominator, 


__       tan  D  (1  —  cos  a)  .      a 

tan  [(D)-  D]  =    —±-1  =  sm  2  D  sin'-. 


Suppose,  now,  a  to  be  expressed  numerically  in  minutes,  and  (D)  —  Dm 
onds;  then 

tan  [(D)  —  D}  =  [(D)  —  D]  sin  1"  ; 

sin  |  =  ~  sin  1'  =  (30  sin  1")  a. 

Therefore,  by  substitution,  we  find 

(D)  —  D  =  (900  sin  1"  sin  2  D)  •*. 
Consequently,  assuming 

F=  90000  sin  1"  sin  2  D  =  [9.63982]  sin  2  Dt 
we  shall  have 


Hie  value  of  Ft  argument  D,  is  contained  in  the  following  small  table 


APPENDIX    XT. 


387 


Factor  F  for  a  correction. 

D 

7? 

D 

F 

D 

F 

o 

o 

0 

0 

•  ooo 

10 

•  149 

20 

•  280 

I 

•oi5 

•      II 

.164 

21 

•  292 

2 

•  o3o 

12 

.178 

22 

.3o3 

3 

•  o46 

i3 

.191 

23 

•  3i4 

4 

•  061 

i4 

•205 

24 

•  324 

5 

•076 

i5 

.218 

25 

.334 

6 

.091 

16 

•23l 

26 

.344 

7 

.106 

17 

.244 

27 

•  353 

8 

•  i  20 

18 

.266 

28 

.362 

9 

•  i35 

'9 

.268 

29 

•  37o 

10 

•149 

20 

•  280 

a  corr.  in  seconds  =  F  .  (—  } 
\io/ 

a  denoting  the  number  of  minutes  it  contains. 

From  what  has  preceded,  it  is  evident  that  a  =  a  —  A  «,  is  the  apparent  dif- 
ference of  the  right  ascensions  of  the  bodies,  and  that  D'  =  D  —  AD  is  the  appa- 
rent declination  of  the  moon ;  and  that 


x  =  [D'  -f  (a  —  A  a)  corr.]  —S   ) 
y  —  [a      —  A  «]  COS  D'  ) 

and  consequently  also 

xl  =  Dl  —  A  Dl 

y\  =  (ai  —  A  ai)  COS  D' 

Moreover,  the  figure  occupying  so  small  a  portion  of  the  sphere,  and  being  com- 
posed of  arcs  of  great  circles,  we  may,  without  any  appreciable  error,  treat  these 
arcs  as  straight  lines  ;  thence  we  shall  obviously  have 


(14) 
(15) 


tan  8=    , 


sin  S      co&S 


Hourly  motion  in  the  orbit  =  -£&-* 
cos « 


Again,  in  the  triangles  B  S  M,  E  S  M, 


(16) 


and  consequently,  by  plane  trigonometry, 
JBM--  -8in 


E  M  =  -51  sin  [M  .-(£+<) 

COS  w 


388  SPHER1JAL   ASTRONOMY. 

With  the  above  hourly  motion  in  the  orbit  we  shall  therefore  have 


Time  of  describing  • 


yi  cos  » 

„_        Wcos  t     . 
n  M= sin 

EM=  Wcost  8in  rw  _ 


Let,  now,  ti,  fa,  be  corrections  to  be  applied  to  the  time  assumed  to  get  the  times 
of  beginning  and  ending,  and  (t)  the  correction  for  the  time  of  the  greatest  phase. 
Then  we  have  evidently 

C    t1    J  CBM\  C  negative  J 

•?    (t)    >  =  the  time  of  describing  In  M  s.  with  a  <  negative  >  sign. 

(    £2    )  {JSM}  (positive) 


To  have  these  times  expressed  in  seconds,  assume 

(w 


yj  cos 
and  then  we  shall  derive 


/1==cein  [—  (8+  0  —  »],  fc 

(f)  =  c  cos  b>  sin  [  —  (S  4-  0  ], 
and  h°ice 

'      beginning     J  (  c  sin  [—  (5  +  t)  —  »]  ) 

The  time  of  <  greatest  phase  >  =  assumed  time  -f-  <  c  cos  w  sin  [  —  (8  +  t)  ]  V      (18) 
(        ending        )  (  c  sin  [—  (8  +  i)  +  »]  ) 

It  has  been  observed,  that  any  one  of  these  values  will  be  the  more  to  be  de- 
pended on  the  more  nearly  it  approximates  to  the  assumed  time.  Thus,  if  the 
assumed  time  be  within  ten  minutes  or  so  of  the  end  of  the  eclipse,  the  point  M 
will  approximate  so  closely  to  the  point  E,  that  no  sensible  error  can  arise  by 
supposing  the  small  portion  ME  of  the  orbit  to  be  a  straight  line,  and  to  be 
passed  over  by  the  moon  with  a  uniform  motion.  This  circumstance  renders  it 
advisable,  in  the  first  instance,  to  take  the  assumed  time  near  to  the  time  of  the 
middle  of  the  eclipse,  so  as  to  give  a  good  result  for  the  time  of  the  greatest  phase, 
and  results  for  the  times  of  beginning  and  ending,  which  may  be  nearly  equally 
relied  on.  Such  a  computation  will  be  sufficiently  exact  for  the  usual  purposes  of 
prediction.  When  the  time  of  beginning  or  ending  is  wanted  to  great  minute- 
ness to  compare  with  observation,  it  will  only  be  necessary  to  repeat  the  operation 
for  ft  time  assumed  as  near  as  convenient  to  the  first  determination,  which  will 
mostly  give  within  a  fractional  part  of  a  second  of  the  true  theoretical  result  ;  a 
degree  of  accuracy,  however,  seldom  wished  for,  and  quite  unsupported  by  the 
present  state  of  the  lunar  theory. 

To  fix  on  a  time  near  to  the  middle  of  the  eclipse  for  the  radical  computation, 
one  of  the  most  simple  expedients  will  be  to  determine  roughly  the  time  of  the 
apparent  conjunction. 


.TY   i 


APPENDIX    XI. 

"We  shall  now  briefly  consider  the  apparent  positions  of  the  moon,  as  related  to 
'.he  sun's  centre. 

It  is  clear  that  S  is  the  angle  of  position  of  the  moon's  centre  from  the  north 
towards  the  east,  at  the  time  assumed ;  also  that  the  angle  N~  S  £  =  u>  -f-  *  is  the 
similar  angle  of  position  from  the  north  towards  the  west  at  the  time  of  begin- 
ning; and  that  the  angle  N S E  =  u> —  t  is  the  angle  of  position  from  the  norih 
towards  the  east  at  the  time  of  ending  ;  and  that  the  angle  N  Sn  =  i  is  the  same 
angle  towards  the  west  at  the  time  of  the  greatest  phase.  Therefore,  by  estima- 
ting all  these  angles  towards  the  east  we  shall  have 

f      beginning     J  f  ( -  «)  -  w  J 

At  ^greatest  phase  >   /  of  J)  's  centre  from  N.  towards  E.  =  j(—  i)          V        (19) 
(        ending        )  ((-  i)  -f-  w) 

In  the  computation  of  the  parallax  in  declination,  we  find  an  angle  M,  which  iu 
practice  may  be  supposed  to  be  the  angle  N 8  Z  for  the  assumed  time,  the  zenith 
Z  being  reckoned  towards  the  east;  consequently,  at  this  time  we  shall  have  S—JM 
for  the  angle  of  position  of  the  moon's  centre  from  the  zenith  towards  the  east. 
At  any  other  time  the  parallactic  angle  J/for  the  latitude  of  Greenwich  may  be 
taken  from  the  following  table,  arguments  the  corresponding  apparent  time  and 
the  sun's  declination.  This  table,  for  any  other  place,  may  be  computed  by  for- 
mulae, such  as  at  page  381,  viz. :  • 

tan  9  =  cot  I  cos  A,  tan  M  = • — ; — r-  tan  A, 

cos  (9  +  <*) 

A  being  the  angle  answering  to  the  apparent  time. 

Those  who  may  be  engaged  in  the  computation  of  eclipses,  for  any  particular 
places,  will  fiud  considerable  facility  in  the  formation  of  similar  tables. 


For  an  occultation  of  a  star  by  the  moon,  the  argument,  instead  of  the  apparent 
time,  will  be  the  star's  hour  angle,  or  the  sidereal  time  minus  the  star's  right  as- 
cension. In  this  case  the  required  positions  will  be  those  of  the  star  with  respect 
to  the  moon's  centre,  which  will  therefore  be  different  from  the  angles  of  position 
for  a  solar  eelipse,  in  which  the  moon's  centre  is  referred  to  that  of  the  sun.  The 
angular  positions  of  the  contacts  at  immersion  and  emersion  will  consequently  be 
determined  in  the  same  way  as  for  an  eclipse  of  the  sun,  and  will  be  estimated  in 
the  opposite  directions.  Thus,  for  an  occultation, 


And  so  must  180C  be  applied  to  the  other  angles  of  position,  as  expressed  for  a 
solar  eclipse  :  this  will  make  the  expressions  for  the  direct  images  of  occultations 
the  same  as  those  for  the  inverted  images  of  eclipses  of  the  sun,  in  estimating  the 
contacts  either  from  the  north  point  or  from  the  vertex. 


390 


SPHERIUAL    ASTRONOMY. 


Parallactic  Angles  for  the  Latitude  of  Greenwich, 

(same  sign  as  /*) 

Arguments  :  Apparent  Hour  Angle  and  Declination. 

Hour  Angle  h. 

Dec. 

North. 

o 

IO 

20 

3o 

4o 

5o 

60 

70 

80 

90 

100 

no 

120 

i3o 

140 

o 

o 

o 

0 

o 

o 

o 

o 

o 

o 

0 

o 

o 

o 

0 

o 

O 

O 

8 

i5 

22 

27 

3i 

35 

37 

38 

39 

38 

37 

35 

3i 

27 

I 

O 

8 

i5 

22 

27 

32 

35 

37 

38 

39 

38 

37 

34 

3i 

27 

2 

o 

8 

16 

22 

28 

32 

35 

37 

38 

89 

38 

37 

34 

3i 

27 

3 

o 

8 

16 

22 

28 

32 

35 

37 

38 

39 

38 

36 

34 

3i 

26 

4 

0 

8 

16 

23 

28 

32 

35 

37 

38 

39 

38 

36 

34 

3i 

26 

5 

o 

9 

16 

23 

28 

33 

36 

38 

39 

39 

38 

36 

34 

3o 

26 

6 

o 

9 

17 

23 

29 

33 

36 

38 

39 

39 

38 

36 

34 

3o 

26 

7 

0 

9 

17 

24 

29 

33 

36 

38 

39 

39 

38 

36 

34 

3o 

26 

8 

0 

9 

17 

24 

29 

34 

36 

38 

39 

39 

38 

36 

33 

3o 

25 

9 

0 

9 

17 

24 

3o 

34 

37 

38 

39 

39 

38 

36 

33 

3o 

25 

10 

o 

9 

18 

25 

3o 

34 

37 

39 

39 

39 

38 

36 

33 

3o 

25 

ii 

0 

9 

18 

25 

3i 

3S 

37 

39 

39 

39 

38 

36 

33 

29 

25 

12 

0 

10 

18 

25 

3i 

35 

38 

39 

4o 

39 

38 

36 

33 

29 

25 

i3 

0 

10 

F9 

26 

3i 

35 

38 

39 

4o 

39 

38 

36 

33 

29 

25 

i4 

0 

10 

J9 

26 

32 

36 

38 

4o 

40 

39 

38 

36 

33 

29 

25 

i5 

o 

JO 

19 

27 

32 

36 

39 

40 

4o 

39 

38 

36 

33 

29 

24 

16 

o 

II 

20 

27 

32 

37 

39 

4o 

4o 

4o 

38 

36 

33 

29 

24 

*7 

o 

1  1 

20 

28 

33 

37 

39 

4o 

4i 

4o 

38 

36 

33 

20 

24 

18 

o 

II 

21 

28 

34 

38 

4o 

4i 

4i 

4o 

38 

36 

33 

29 

24 

'9 

o 

11 

21 

29 

34 

38 

4o 

4i 

4i 

4o 

38 

36 

33 

29 

24 

20 

o 

12 

22 

29 

35 

39 

4i 

4.i 

4i 

4o 

38 

36 

33 

29 

24 

21 

0 

12 

22 

3o 

36 

39 

4i 

42 

42 

4o 

39 

36 

33 

29 

24 

22 

0 

12 

23 

3o 

36 

4o 

42 

42 

42 

4i 

39 

36 

33 

29 

24 

23 

0 

13 

23   3l 

37 

4o 

42 

43 

42 

4i 

39 

36 

33 

2.9 

24 

24 

0 

r3 

24  32 

38 

4i 

43 

43 

42 

4i 

39 

36 

33 

29 

24 

25 

o 

i4 

25  j  33 

38 

42 

43 

43 

43 

4i 

39 

36 

33 

29 

24 

26 

0 

i4 

36  i  34 

39 

42 

44 

44 

43 

42 

39 

36 

33 

29 

24 

27 

0 

i4  26  35 

4o 

43 

44 

44 

43 

42 

39 

36 

33 

29 

24 

28 

o 

i5 

27  35 

4i 

43 

45 

45 

44 

42 

4o 

37 

33 

29 

24 

29 

o 

if 

28 

36 

4i 

44 

45 

45 

44 

42 

4o 

37 

33 

29 

24 

By  subtracting  the  parallactic  angle,  for  the  respective  times  of  beginning, 
greatest  phase,  and  ending,  from  the  foregoing  angles  of  position  of  the  moon'.s 
centre  from  the  north  towards  the  easx,,  wo  shall  evidently  obtain  the  same  angle* 
from  the  zenith  or  vertex  towards  the  east. 

If,  however,  the  operation  be  repeated  for  the  accurate  determination  of  the 
times  of  .beginning  and  ending,  we  shall  have  in  the  calculations  the  angle  Jfalso 
at  thf-se  times.  Let  «j,  t*i,  Mi  be  the  angles  appertaining  to  the  beginning,  and 
12,  «i»2,  J/2  those  for  the  ending,  and  we  shall  evidently  have  the  following  values, 
which  will  be  more  accurate  than  the  preceding : 


APPENDIX    XI. 


391 


Parallactic  Angles  for  the  Latitude  of  Greenwich. 

(game  sign  as  Ji) 

Arguments  :  -Apparent  If  our  Angle  and  Declination. 

Hour  Angle  h. 

Dec. 

South. 

o 

10 

20 

3o 

4o 

5o 

60 

70 

80 

90 

100 

110 

120 

i3o 

140 

0 

o 

8 

i5 

22 

27 

3i 

35 

37 

38 

39 

38 

37 

35 

3i 

27 

i 

0 

8 

i5 

21 

27 

3i 

34 

37 

38 

39 

38 

37 

35 

32 

27 

2 

o 

8 

i5 

21 

27 

3i 

34 

37 

38 

39 

38 

37 

35 

32 

28 

3 

o 

8 

i5 

21 

26 

3i 

34 

36 

38 

39 

38 

37 

35 

32 

28 

4 

o 

7 

i5 

21 

26 

3t 

34 

36 

38 

39 

38 

37 

35 

32 

28 

5 

o 

7 

i5 

21 

26 

3o 

34 

36 

38 

39 

39 

38 

36 

33 

28 

6 

0 

7 

14 

20 

26 

3o 

34 

36 

38 

39 

39 

38 

36 

33 

29 

7 

o 

7 

«4 

2O 

26 

3o 

34 

36 

38 

39 

39 

38 

36 

33 

29 

8 

o 

7 

14 

2O 

25 

3o 

33 

36 

38 

39 

39 

38 

36 

34 

29 

9 

o 

7 

M 

2O 

25 

3o 

33 

36 

38 

39 

39 

38 

37 

34 

3o 

10 

0 

7 

M 

2O 

25 

3o 

33 

36 

38 

39 

39 

39 

37 

34 

3o 

ii 

o 

7 

M 

20 

25 

29 

33 

36 

38 

39 

39 

39 

37 

35 

3i 

12 

o 

7 

U 

20 

25 

29 

33 

36 

38 

39 

4o 

39 

38 

35 

3i 

i3 

0 

7 

'4 

J9 

25 

29 

33 

36 

38 

39 

4o 

39 

38 

35 

3i 

i4 

0 

7 

i3 

*9 

25 

29 

33 

36 

38 

39 

4o 

4o 

38 

36 

32 

i5 

o 

7 

i3 

J9 

24 

29 

33 

36 

38 

39 

4o 

4o 

39 

36 

32 

16 

0 

7 

i3 

'9 

24 

29 

33 

36 

38 

4o 

4o 

4o 

39 

37 

32 

17 

o 

7 

i3 

r9 

24 

29 

33 

36 

38 

4o 

4i 

4o 

39 

37' 

33 

18 

o 

7 

t3 

'9 

24 

29 

33 

36 

38 

4o 

4i 

4i 

4o 

38 

34 

T9 

o 

7 

i3 

!9 

24 

29 

33 

36 

38 

4o 

41 

4i 

4o 

38 

34 

20 

o 

7 

i3 

'9 

24 

29 

33 

36 

38 

4o 

4i  14i 

4i 

39 

35 

21 

o 

6 

i3 

'9 

24 

29 

33 

36 

39 

4o 

42 

42 

4i 

39 

36 

22 

o 

6 

i3 

X9 

24 

29 

33 

36 

39 

4i 

42 

42 

42 

4o 

36 

23 

o 

6 

i3 

18 

24 

29 

33 

'36 

39 

4i 

42 

43 

42 

4o 

37 

24 

o 

6 

i3 

18 

24 

29 

'33 

36 

39 

4i 

42 

43 

43 

4i 

38 

25 

o 

6 

i3 

18 

24 

29 

33 

36 

39 

4i 

43 

43 

43 

42 

38 

26 

o 

6 

i3 

18 

24 

29 

33 

36 

39 

42 

43 

44 

44 

42 

39 

27 

0 

6 

i3 

18 

24 

29 

33 

36 

39 

42 

43 

44 

44 

43 

4o 

28 

o 

6 

12 

18 

24 

29 

33 

37 

4o 

42 

44 

45 

45 

43 

4i 

29 

0 

6 

12 

18 

24 

29 

33 

37 

4o 

42 

44 

45 

45 

44 

4i 

(  beginning 
atest  ph 
ending 


For  •<  greatest  phase  f  /  of  J)  's  centre  from  K  towards  E.  =   -j  ( —  » ) 


;<-«»)— .-jr.) 

of  D 's  centre  from  vertex  towards  E.  =  <  ( —  i  )  —  M 


(20) 


These  angles  relate  to  the  natural  appearance  or  direct  images  of  the  bodies. 
For  the  same  angles,  as  they  will  appear  through  ar.  inverting  telescope,  ±  180° 
must  be  applied  :  this  may  be  simply  done  by  using  (180°  — i)  instead  of  ( —  i). 


SPHERICAL    ASTRONOMY. 

To  find  the  time  when  the  apparent  conjunction  takes  place,  let  t  denote  the 
interval,  in  units  of  an  hour,  to  be  applied  to  the  time  of  the  true  conjunction,  and 
h  the  common  hour  angle  of  the  bodies  at  the  true  conjunction.  Then  the 
position  of  the  sun,  not  being  supposed  to  be  influenced  by  parallax,  the  common 
apparent  hour  angle  of  the  bodies,  at  the  time  of  the  apparent  conjunction,  will 
be  h1  =  h  -f  15° .  t  •  and  therefore  at  this  time, 


sin  (*  +  15°.  t), 
so  that  the  conditi  n  for  apparent  conjunction,  viz.  a'  =  a  —  A  a  =  0,  gives 


for  the  determination  of  the  interval  t,  which  from  this  equation  will  be  best  found 
perhaps,  by  the  usual  method  of  double  position.  We  only  want,  however,  an  ap- 
proximate value,  and  may  therefore  avoid  much  unnecessary  labor  in  estimating 
this  time.  Thus,  at  the  time  of  true  conjunction,  the  same  approximate  formulae 
may  be  adopted  as  used  at  page  385,  viz.  • 

_,,  cos  I     , 

A  a  =  P  -  -  sin  ft, 
cos  D 


A*,=P'^sinl")^}cosA, 
\dt  /  cos  D 


in  which  —  applies  to  the  moon.     It  is  evident,  then,  as  the  true  positions  of  the 

bodies  have  no  difference  of  right  ascension,  that  A  a  is  the  apparent  difference  of 
right  ascension  ;  and  consequently,  as  the  relative  apparent  motion  in  right  as- 
cension is  ai  —  A  <»i  or  ai  —  P'  (—-  sin  1")  — —  w>s  A,  the  correction  t  to  be 
\d  t  /  cos  D 

applied  to  the  time  of  true  conjunction  to  get  that  of  the  apparent,  will  be 

rv     COS  I 

P' sin  h 

cos  D  sm  h 

t  = 


/dh            \  coal                         cos  D  /dh    .        \ 

t—P'  I  —  sin  1") cos  h       ai  -Dr :  —  (77  sin  1")   cos  h 

\dt  /  cos  D  P   cos  I         \dt  / 

the  calculation  of  this  expression,  we  mi\y  us 
as  a  mean  value  of  D.     Assume,  therefore, 

100  cos  D  _100     cos  U°  _  [0.23103] 
'  ~    P1  cos  I    '~  ~57~  "    cos  /  cos  I 


To  facilitate  the  calculation  of  this  expression,  we  may  use  57'  as  a  mean  value 
for  P'  and  14°  as  a  mean  value  of  D.     Assume,  therefore, 


6  =  100  (^  sin  1")  cos  h  =  [1.40274]  cos  h 

gO=  100  sin  A 
for  which  the  nearest  whole  numbers  will  suffice,  and  we  shall  have 


The  values  of  the  factor/  are  given  for  various  principal  places  in  the  table  at 
page  406  :  for  any  place  not  contained  in  that  table  it  can  be  computed  from  the 
above  expression,  and  used  as  a  constant  factor  for  all  eclipses  at  that  place.  The 
Talues  of  tf,  6(l\  are  also  tabulated  at  page  4')5,  where,  f  >r  convenience,  the  argu- 
zuent  h  Is  given  in  time. 


APPENDIX    XI.  393 

II. — FORMULAE  OF  REDUCTION  TO  DIFFERENT  PLACES. 

Before  quil  ing  this  subject  we  shall  give  a  method  of  calculating  numerical 
equations  which  will  serv.e  to  determine,  with  much  ease  and  with  sufficient  accu- 
racy, the  circumstances  of  an  eclipse  of  the  sun  for  any  place  comprised  within  a 
certain  range  of  country.  To  effect  this  purpose  in  the  most  ample  manner,  in 
again  proceeding  with  the  general  determination  of  the  time  of  a  phase,  whose 
apparent  distance  of  centres  is  A',  we  shall,  in  the  expressions,  separate  as  much 
as  possible  the  quantities  whi-ch  involve  the  position  of  the  place  on  the  earth. 
The  values  of  the  co-ordinates  x,  y,  given  at  p.  387,  observing  that  a  —  A  a  =«', 
may  be  put  down  as  follows: 

x  =  [(D  -f  a'  corr.)  -  <J]  —  A  D, 
y  —  a.  cos  D'  —  A  a  cos  D', 

and  will  thus  consist  of  two  terms,  over  the  former  of  which  the  particular  place 
on  the  earth  has  but  little  influence.  If  i  denote,  as  before,  the  inclination  of  the 
apparent  relative  orbit,  these  ordinates  resolved  in  the  direction  of  n,  perpendic 
ular  to  the  orbit,  and  in  the  direction  of  the  orbit,  will  give  x  cos  «  —  y  sin  i,  aud 
x  sin  r-f-  y  cos  «.  It  is  evident,  (hen,  that  x  cos  »  —  y  bin  t  represents  n,  the  near- 
est approach,  and  x  sin  «  +  y  coe  <  the  distance  of  the  moon  from  it,  which  distance 
is  estimated  in  the  direction  of  her  motion.  At  the  time  of  the  beginning  or  end- 
ing of  the  phase,  the  distance  of  the  moon  past  the  nearest  approach,  or  greatest 
phase,  will  be  ^  A '  sin  <o ;  therefore  the  moon  precedes  this  position  by  a  dis- 
tance c-niial  to  ^  A'  sin  u  —  (x  sin  t  -}-  y  cos  <),  which,  divided  by  -^-,  the  hourly 

COS  ' 

motion  in  the  orbit,  gives  ^p  — sin  w (x  sin  <  +  y  cos  t)  for  the  inter- 

y\  y\ 

val,  in  units  of  an  hour,  to  be  applied  to  (he  assumed  time  Tto  get  the  time  t 
when  the  phase  takes  place.  Assume,  therefore, 

k  =  [3.65630]—— ,    .    .    (1) 

and,  the  time  being  counted  in  seconds, 

t  =  T  =f  k  sin  w ;  (a:  sin  «  +  #  c°s  0 (2) 

Also,  x  cos  i  —  y  sin  <  expressing  the  nearest  approach,  we  evidently  hare 

x  cos  t  —  y  sin  t 
cosu>  = ^ (3) 

Make  now  the  following  assumptions : 

(D  +  a'  corr.)  —  $  a  cos  D1 


~          .          A'  A' 

k  k 

=  — ;  [(D  -f  a'  corr.)  —  6]  sin  t  -\ 7  a  cos  D'  cos  « 

A  A 


(4) 


AD                   A  a  cos  D'    .                    •] 

k                          k 
A  q  =  —  ;  A  D  sin  i  -\  ;  A  a  cos  D'  cos  t 

A                             A 

and>  observing  the  above  values  of  a-  and  y,  the  equations  (2),  (3)  will  become 

-<,- (6) 


394  SPHERICAL  ASTRONOMY. 

Let  y,  \L  be  determined  by  the  equations 


._    ,   _  corr)  —  a' 

y  COS  ^  =  ^ ; '— 

a  COS  D' 
y  Sin  w  = ; 

A 

and/>,  q  will  take  the  following  values: 

p  =    y  cos  O//  -f  t)  ) 


(7) 


It  yet  remains  to  determine  the  values  of  A  p,  A  g,  which  depend  on  the  po- 
sition of  the  place  of  observation.  Adopting  the  notation  used  in  the  equations 
(3),  (4),  (9),  (10),  pages  379  and  382,  we  shall  have 

[5.31439]  A     cos  I     . 

A  a  =  ±—      -J—  . 7:  sin  A, 

1  —  n         cos  U 

[5.31439]  A  \  .                                 .     n  cos  (A  +  i  A  a)l 
A  D  =  fc —  I  sm  J  cos  D  —  cos  /  sm  D s — \ I. 

1  —  n!       L  cos  i  A  a      J 

To  simplify  the  expressions,  let 

_  [5.31439]  A    cos  D' 
~~  (I— w)  A'  '  cOsZ>' 

[5.31439]  A  [5.31439]  A 

c  =  L J_  .  cos  I>,  a  =  7T—  ""  •  am  2) 

(1  —  HI)  A'  (1 

ftnd 

b  A'  cos  /  sin  h 

A  a  = j^ , 

cos  D 

cos  (A  +  i  A  «) 


A  I>  =  c  A '  sin  /  —  a  A'  cos  Z 


cos  i  A  a 


=  c  A '  sin  /  —  a  A '  cos  £  cos  h  -f-  a  A'  tan  -—  cos  /  sin  A. 
These  substituted  in  (5)  give 
A  p  =     c  cos  i  sin  /  —  cos  l\     a  cos  «  cos  A  —  (   a  cos  t  tan  — —  —    6  sin  t)  sin  A  I 

A  q  =  A;  c  sin  t  sin  Z  —  cos  /  I  A;  a  sin  t  cos  A  —  (k  a  sin  i  tan  — \-kb  cos  i)  sin  A  I 

The  value  of  b  contains  the  factor =-.  for  which  we  have 

cos  D 

--^?-  =  cos  A  D  (1  +  tan  D  tan  A  D). 
cos  JJ 

Substitute  the  first  value  of  tan  A  Dt  p.  379,  and  ' 

1  —  p  sin  P ^  cos  (A) 

cos  D'  cos  D  '      v  ' 


=  cos  A  D  . 


1  — p  sin  P  [sin  /  sin  D  +  cos  /  cos  D  cos  (A)]' 

Or,  putting  A  instead  of  (A)  in  the  numerator,  which  cannot  sensibly  affect  the 
value  of  the  fraction, 

cos  D'  1  —  n 

f-  —  cos  A  D  . . 

cos  D  1  —  H! 


APPENDIX    XI.  395 

This,  supnosing  cos  A  D  =  1,  reduces  the  values  of  the  constants  a,  b,  c,  to  the 
following : 

[5.31439]  A  1 

-(i^Srz7         I  .  .(9) 


If  e  be  a  small  arc  determined  by  g  cos  e  =  b,  g  sin  e  =  a  tan  -  ,  we  shall  have 
a  cos  i  tan  —  --  b  sin  i  =  </  sin  (  —  i  -f  «)  =  <?  cos  (90°  +  i  —  e)  ; 

A;  a  sin  i  tan  --  1-  k  b  cos  t  =  kg  cos  (<  —  e)  =  &  <y  sin  (90°  -j-  *  —  «)• 

However,  as  e  must  always  be  a  very  small  arc,  we  may  suppose  cos  e  =  1 
also  g  =  b,  and,  «  being  expressed  in  minutes, 


If  therefore 

x  =  (90°  +  0  —  «      ......  .  •    •    •  (") 

the  values  of  A  p,  A  <?,  will  be 

A  p  =  c  cos  i  sin  J  —  cos  /  (a  cos  t  cos  h  —  6  cos  x  sin  h) 
A  y  =  k  c  sin  t  sin  I  —  cos  I  (k  a  sin  i  cos  A  —  k  b  sin  x  sin  ^)  ' 
Assume  now 

X  =  the  longitude  of  the  place,  -f"  ea8*»  —  west. 
#=  the  true  hour  angle  of  the  moon,  for  the  meridian  of  Greenwich. 

L1  =  c  cos  t  \ 
y'  cos(<//  —  #)  =  acos«   I     ........  (18) 

y'  sin  (>//  —  H)  =  6  cos  x  * 

i"  =  fc  c  sin  t    \ 
y"  co&W  —  H)  =  kasint   1.     .......  (14) 

y"  sin  («//'  —  ^T)  =/fc6sinx  ' 
and  we  shall  have 

Ap  =  L'  sinl  —  yf  cos  J  cos  (V  +h  —  H)=.  L'  sin  /  —  y'  cos  /  cos  ty'  +  A), 
&q  =  L"  sin  /  —  y"  cos  /  cos  ($"  +  h  —  H)=zL"  sin  I  —  y"  cos  /  cos  (<£"  +  A)> 
so  that  the  equations  (6)  will  become 

cos  u  =  p  —  Lf  sin  ^  +  y'  cos  /  cos  (^'  +  X) 


=  (T7-  ?)  T  Ar  sin  u)  -f-  i"  sin  /  -  y"  cos  J  cos  ty"  +  X)  ' 
After  computing  the  constants  k,  p,  q,  L't  L",  »//',  ^",  by  means  of  the  equations 
(1).  (H  (8)»  (9),  (10),  (11),  (13),  and  (14).  we  shall  thus  have  two  numerical  equa- 
tions for  the  determination  of  o>  and  the  Greenwich  time  t  of  the  phase,  for  any 
place  whose  latitude  is  /  and  longitude  X.  The  accuracy  of  the  determination  will 
principally  depend  on  the  proximity  of  the  resulting  time  t  to  the  assumed  time 
T;  and  therefore  the  result  will  be  near  the  truth  for  all  places  where  the  phase 
will  take  place  near  to  this  time. 

In  making  these  calculations  for  any  particular  portion  of  country,  which  for  the 
partial  |>hase  will  be  necessary  for  lioth  the  beginning  «hd  ending,  it  will  be  best 
in  the  first  instance  to  fix  upon  a  place  near  the  centre  ui.d  compute  the  eclipse 
for  that  place,  which  computation  will  furnish  good  .mean  values  for  the  data  Dt 
t,  ft,  a'  corr,  A  D,  A  a,  i,  yi,  A',  A,  and  comp.  log  (1  —  «,). 


396  SPHERICAL    ASTRONOMY. 

f  cos  V  -  y',  f  cos  I'1  =  y" 

By  suppo^g  '  ^  f  I  *  'x,  J,  s.u  ,„  =  I  L         .    .    .     .  (It) 

the  expressions 

—  U  Bin  I  -f-  y'  cos  I  cos  ((//•'  -f"  A), 

—  L"  sin  J  -f  y"  cos  /  cos  (t//"  -f  X), 
will  take  the  forms 

('  [sin  /'  sin  /  -f-  cos  /'  cos  I  cos  (V  -f  A)], 
{"  [sin  f"  sin  I  +  cos  J"  cos  /  cos  (<//'  4-  A)]  ; 

and,  without  the  factors  £',  £",  will  represent  the  cosines  of  the  distances  of  the 
proposed  place  from  two  other  places  whose  latitudes  are  /',  /'',  and  west  longi- 
tudes i//',  !//".  The  former  of  these  two  places  will  be  near  to  the  southern  pole  of 
the  truH  relative  orbit,  and  the  latter  will  be  near  to  the  orbit  itself,  and  will  pre- 
cede the  moon  by  a  distance  nearly  equal  to  90°. 

For  purposes  which  do  not  require  gr^at  minuteness,  the  preceding  equations 
will  admit  of  some  simplification  by  neglecting  the  small  angle  e.  Add  the 
squares  of  the  equations  (13)  and  (14),  observing  that  c2  -f-  a2  =  b\  and 

i'2  -f  y'3  =      62  (cos2  1  4-  cos2  x), 
,-  X"2  +  y"2  =  £2  62  (sin2  i  +  sin2  x)  ; 

which  give  the  general  relation 

£'a  +  y''  +  ^2  +  ^a  =  2&2     ......  (17) 

By  neglecting  0,  x  =  90°  -f-  1,  cos  x  ==  —  sin  «,  sin  ^  =  cos  i  ;  and  then 


which  united  with  the  equations  (16)  give  f  =  b,  f  =  k  bt  and  hence 

L'  c  cos  i  _. 

in  I'  =  —  -r  =  --  7  —  =  -  cos  D  cos  i  ; 


sin 


•=.  ('  cos  I'  =  b  cos  /'  ; 
6  sin  i 


•    /,'       IT\ 
•mfy  -H)  =  —  —  =  - 


'  =  {''  cos  /"  =  kb  cos  /"; 

.        k b  sin  x       kb  cos  <        cos  i 


sin  I'  =  —  cos  2>  cos  t 
X'  =  —  b  sin  /' ;  y'  -  6  cos  /' 


(18) 


ein  V  =  —  cos  J)  sin  c 
=  — A  6  sin  I"  ;  y"  =  ko  c-us  t     i  ^  /j^-v 

«U(^-^)  =  ^L, 
v*  '       cos  I" 


APPENDIX   XI.  397 

These  may  be  employed  instead  of  the  equations  (13)  and  (14) ;  or  the  equations 
(18)  and  (14)  may  be  adopted  in  their  reduced  form,  viz. : 

—  =  cos  D  cos  i 

o 

•j-  cos  (i//  —  H)=.  sin  D  cos  c  ••..,,.    (20) 

— -  sin  (if/'  —  H)  =  —  sin  t 
6 

T " 

- —  =  cos  D  sin  « 

k  b 

^7  cos  (;//"  —  H)  =  sin  D  sin  « (21) 

j-r  sin  ($"  —  H)  =  cos  i 
in  which  the  coefficients  c,  a,  will  not  be  required. 


III. — TRANSITS  OF  MERCURY  AND  VENUS  OVER  THE  DISK  OF  THE  SUN. 

These  phenomena  are,  in  many  respects,  analogous  to  that  of  an  annular  eclipse 
of  the  sun,  and  admit  of  a  similar  calculation  ;  the  principal  distinction  consists  in 
the  negative  sign  of  the  relative  motion  of  the  planet  in  right  ascension,  which 
will  make  the  inclination  of  the  orbit  always  obtuse,  and  therefore  render  some 
modifications  necessary  in  the  determination  of  the  particular  species  of  the  other 
angles  which  enter  into  the  computation.  To  avoid  any  confusion  that  might 
thus  arise,  we  shall  adopt  the  sun  as  the  movable  body,  and  refer  his  positions  to 
that  of  the  planet  which  we  now  suppose  to  be  stationary.  Thus, 

6  =  the  O's  declination; 
])  =  the  planet's  declination  ; 
it  =  the  O's  equatorial  horizontal  parallax, 
P  =  the  planet's  equatorial  horizontal  parallax; 
«  =  O's  right  ascension  minus  that  of  the  planet; 
x  —  (V  -f  a'  corr.)  —  D  ; 
y  =  a'  cos  i' ; 

a-j  =  the.  O's  motion  in  declination  minus  that  of  the  planet ; 
y\  =  (O's  motion  in  right  ascension  minus  that  of  planet) .  cos  6' ; 

and  so  we  might  proceed  as  with  nn  eclipse  of  the  sun,  only  observing  that  the 
relative  parallax  p  («  —  P)  is  a  negative  quantity,  and  that  the  positions  of  the 
contacts  on  the  limb  of  the  sun,  as  in  the  case  of  an  occultation,  will  be  at  points 
opposite  to  those  which  come  out  in  the  calculation.  However,  as  the  relative 
parallax  is  always  very  small,  the  ingress  and  egress  of  the  planet  will  be  seen  at 
all  places  on  the  earth  at  nearly  the  same  absolute  time  ;  it  will,  for  this  reason, 
be  best  to  compute  first  the  circumstances  for  the  centre  of  the  earth,  and  then  to 
ascertain  the  small  variations  produced  by  parallax  for  any  assumed  place  on  the 
surface,  which  may  be  readily  deduced  from  the  preceding  equations  for  the  reduc- 
tion of  an  eclipse  of  the  sun.  Let  w,  (t),  be  the  values  of  »,  t,  for  the  centre  of  the 
earth,  and,  by  separating  the  effects  of  parallax  from  the  equations  (6), 


398 


SPHERICAL    ASTRONOMY. 
.£.  cos  w  =  p, 


A  cos  w  =  A  p, 


A  t  =  —  A  q  T  k  A  sin  w. 


But,  as  the  quantities  A  cos  w,  A  sin  w  are  very  small,  A  sin  w  =—  A  cos  w  -  - 

sin  w 

that  is,  A  sin  w  =  —  A  »  -  -  .     Therefore, 
r  sin  w 

cos  w  /,         cos  w  \ 

A  t  =  —  A  q  ±k  Ap  -.  -  =  ±  (k  Ap-  -  T  A  q). 
sin  w  \  sin  w  / 

In  this  expression  substitute  the  values  of  A  p,  A  q,  according  to  the  equations 
(12),  and  we  find  A  t  = 

.r.    cosf—  iTw]..  /       cos[—  tTw]  ,  Tco8[~—  xTwl    .     7\1 

±1  ke  -  K  ---  ±  sin  I—  cos  ilk  a  -  ±  -  Jcosh-kb  -  L.  J  sin  A)  |, 

L  sin  w  \  sin  w  sin  w  /  J 


?-&-— 


-,  c  =  6  ccs  5  and  a  =  6  sin  a. 


in  which  b  = 

A  A 

Because  of  the  smallness  of  the  parallax,  the  angle  ewill  not  be  appreciable,  ami 
consequently  %  =  90°  -f  *,  cos  [  —  x  T  w]  =  sin  [  —  t  qp  w].  We  shall  therefore 
have  for  the  time  of  ingress  or  egress  the  following  general  expression,  in  which 
the  terms  within  the  brackets  depend  on  the  position  of  the  place  of  observation  ; 
also  the  upper  signs  apply  to  the  ingress,  and  the  UT^aer  signs  to  the  egress. 

/  =  T—  q  T  k  sin  w 


.cosf  —  «Tw]  .         /.      cosf—  tTw]  8in[--iqFTr]  .    .\        ,1 

cos<5  -  ^  --  -sin/—  |sm<$  —  =;  ---  -cosh  --  ^  —  -  --  isuaAlcog/  I 
sin  w  \  sin  w  sin  ,v  /         J 

Assuming  k"  =  —  :  -  ,  this  expression  will  resolve  into  the  following  : 
p  sin  w 


A  °08  ' 


(a  -f  «  corr.)  —  D 

y  cos  $  =  ^-± '- 

A 

y  sin  i//  =  — 


COS  W  =:       y  COS 


v  =  k . 

L" 


(U/  -j"  *) 


-} 


(*) 
<<) 


A  sin  w 


~  =  cos[(-0  T  w]  cos  * 

y" 

cos  &"  —  H)  =  cos  [  (-  i)  T  w]  sin  t 


(<*) 


APPENDIX    XI.  399 

In  these  equations, 

H=  the  Q's  true  hour  angle  from  the  meridian  of  Greenwich,  at  the  time  (t). 

For   \  extei:ior  I  contact  of  limbs,  A  =  \  '  +  S  I 
(  interior  }  (  a  —  s  } 

For  contact  of  centre  of  planet  with  ©'s  limb,  A  =  a ', 
s  denoting  the  true  semi-diameter  of  the  planet,  and  <r  that  of  the  sun. 

The  equations  (a),  (6),  (c),  (d)  will  serve  to  determine  the  constants  (t),  y ',  L", 
«//",  for  the  times  of  ingress  and  egress,  and  then  there  will  result  two  numerical 
equations  of  the  form  (e)  to  reduce  the  phenomena  to  any  place  on  the  earth's 
surface. 

For  the  points  on  the  limb  of  the  sun,  we  shall  have 

At  \  ingress  I ,  angle  from  N.  towards  E.  =  \  (ISQ°0  "  '}  "  w  I  for  direct  image.  . 
(  egress    )  (  (180    —  <)  +  w  ) 

or  \  ^~  *'  ~~  W  (•  for  inverted  image. 

<(-  ')  +  w  > 
which  will  be  sufficiently  accurate  for  all  places  on  the  earth. 

The  time  Tma.y  be  assumed  near  to  the  time  of  conjunction  in  longitude,  or 
right  ascension,  as  it  may  suit  convenience.  For  Mercury,  if  very  minute  accuracy 
is  wanted,  it  may  be  necessary,  for  more  correct  values  of  (t),  to  assume  two  times 
T  near  to  the  times  of  ingress  and  egress ;  but  it  is  very  questionable  whether 
such  a  precarious  extent  of  accuracy  would  sufficiently  recompense  the  time  ex- 
pended on  the  calculation. 

IV. OCCTJLTATIONS  OF  STARS  BY  THE  MOON. 

These  may  be  calculated  in  the  same  manner  as  eclipses  of  the  sun,  the  only 
difference  in  the  operation  consisting  in  the  star  having  neither  motion,  parallax, 
nor  semi-diameter.  But  where  great  minuteness  is  not  wanted,  these  particular 
circumstances  will  afford  some  degree  of  simplification  to  the  expressions,  if  that 
parallax  of  the  moon  be  adopted  which  would  answer  to  the  star  as  an  apparent 
place,  since  this  parallax,  at  the  times  of  immersion  and  emersion,  will  then  be 
precisely  that  of  the  respective  points  of  the  moon's  limb  which  come  in  contact 
with  the  star;  and  thus  the  augmentation  of  the  moon's  Sfmi-dinmeter  will  be 
evaded,  so  that  the  true  semi-diameter  may  be  employed.  For  this  novel  and  ju- 
dicious expedient  we  are  indebted  to  Carlini. — See  Zach's  Correspondance,  vol. 
xviii.,  pnge  528. 

As  in  the  case  of  the  pun,  let  &  denote  the  declination,  and  h  the  hour  angle  of 
the  star,  and  let,  P  represent  the  equatorial  horizontal  parallax  of  the  moon. 
Then,  for  the  effects  of  parallax  in  right  ascension  and  declination,  we  must  sub- 
stitute S  for  D't  and  h  for  h  in  the  formulae  (2)  at  p.  379,  which  thus  become,  dis- 
regarding £  A  A, 

,,  cos  I    . 
A  o  =  p  P =•  sin  h, 

008  D 

A  D  =  p  P  (sin  I  cos  S  —  cos  I  sin  &  cos  7i). 

As  soon  as  the  immersion  takes  place,  these  expressions  will  represent  the  parallax 
of  that  point  of  the  moon's  limb  which  is  in  contact  with  the  star;  and  therefore 
the  application  of  this  parallax  to  the  centre  of  the  moon  will  produce  nn  apparent 
distance  A'  of  the  centres,  equal  to  the  true  semi-diameter  s  of  the  moon.  Also 
as  the  star,  in  the  course  of  the  occultation,  is  only  affected  with  its  apparent  diur- 
nal motion,  the  hourly  variations  of  the  above  values  will  be 


400 


dh. 


SPHEE-ICAL    ASTRONOMY. 

_  /dh   .      ,,\  cos  I 

A  ai  =  n  P  I  — -  sin  1    I cos  h. 

\dt  /  cos  D 

A  Di=pP  (  —  sin  1"  1  cos  /  sin  6  sin  7* ; 


in  which  —  is  15°  2'  28",  the  hourly  diurnal  motion  of  the  earth,  and  therefor* 


—  sin  1"  =  [9.41916]. 
Assume 


7.  cos  T 

=  peos/=    ,          — , 


(2)  =  p  sin  1  = 


(1—  <?2)sin  I' 


0(3)  =  p  Cos  I  —  sin  1"  =  [9.41916]  00) 
which  are  constant  coefficients  depending  on  the  latitude  of  the  place ;  then 


A  a=—  - 


sin  A, 


A  ai  = pr  COS  h. 

cos  D 


(2) 


cos  D 

A  D  =  (0(2>  cos  5  —  0O  sin  3  cos  /<) .  P,  ^  J)1=.  0(3) .  P  sin  J  sin  A. 

If,  in  the  values  of  A  a,  A  ai,  we  use  cos  6  instead  of  cos  D,  the  values  of  x,  y,  xlt 
yi,  p.  387,  will  become 

x  =  (D  —  3)  —  (0(2> .  P  cos  t  —  00) .  P  sin  <5  cos  A) 

y  =a  cos  6    —  0'1' .  P  sin  h 

%i  =  Di  —  0t3) .  P  sin  6  sin  A 

yj  =  cti  COS  5    —  0O  .  P  COS  h 

in  which  we  have  disregarded  the  a  correction. 

With  the  values  of  *,  y,  xi,  yi,  so  found,  we  may  then  proceed  with  the  equa- 
tions (16)  and  (18),  pages  387  and  388,  as  in  the  case  of  a  solar  eclipse. 

This  method  is  similar,  and,  as  far  as  accuracy  goes,  the  same  as  the  recent 
method  of  Professor  Bessel,  who  divides  all  the  quantities  by  the  equatorial  hori- 
zontal parallax  of  the  moon.  He  assumes 


P  = 


cos  £ 


(3) 


u  •=•  00)  sin  h,  u'  =  0(3)  cos  h  ) 

v  =  0'2)  cos  &  —  00)  sin  3  cos  h,        v'  =  0'3)  sin  5  sin  h  ) 
so  that  if  we  change  the  signification  of  the  symbols  x,  y,  xi}  yi,  and  suppose  them 
now  to  represent  the  preceding  values  divided  by  P,  we  shall  have 
x=.q  —  v,  Xi  =  q'  —  v' 

v  =  n  —  u,  t/i  =  p'  —  U 


(*) 


(6) 


Tbeae  values  being  adopted,  in  proceeding  with  the  equations  (16)  and  (18)  we 
must  use  A'  ==  — ,  the  value  of  which,  according  tr  Burckhardt's  Tables  de  la  Lune 

(Paris,  1812),  p.  73,  is  [9.43637].     Much  facility  is  thus  given  to  the  calculation 
of  occultations,  for  different  plnces,  if  the  values  of  p,  q,  p',  q',  which  are  indepen 


APPENDIX    XI. 


401 


dent  of  geographical  position,  are  published;  but  if  these  quantities  are  to  be  pre- 
pared by  the  computer,  the  equations  (2)  will  be  more  simple  and  advantageous. 

The  chief  difficulty  in  the  calculation  of  occulta- 
tion?,  for  any  particular  place,  rests  in  the  selection 
of  the  list  of  stars :  in  the  course  of  any  year  a  great 
number  will  be  liable  to  occupation  on  the  earth 
generally,  though  the  majority  of  them  will  not  be 
occulted  at  the  particular  place  for  which  the  special 
calculations  are  to  be  made.  It  will  therefore  be 
expedient  to  reject  such  stars  as  may  at  different 
stages  of  the  calculation  be  shown  to  violate  any 
conditions  necessary  for  the  existence  of  the  occulta- 
tion,  its  appearance  above  the  horizon,  or  its  exemp- 
tion from  the  glare  of  sun-light.  For  the  general 
list  we  may  observe,  that  the  difference  of  declina- 
tion at  the  time  of  conjunction  must  be  within  the 
limit  of  about  1°  30',  and  that  all  stars,  whose  con- 
junctions with  the  rnoon  occur  within  two  days  of 
new  moon,  may  be  omitted.  In  the  process  of  exclu- 
sion for  the  particular  place,  the  first  and  most  pal- 
pable condition  is,  that  at  the  time  of  conjunction 
the  sun  must  be  below,  or  near  to,  the  horizon ;  if 
more  than  half  an  hour  above  the  horizon,  the  occul- 
tation  will  surely  be  useless;  another  condition  is, 
that  the  star  must  be  above  the  horizon;  and,  to 
satisfy  this,  the  hour  angles  at  the  times  of  immer- 
sion and  emersion  must  be  less  than  its  semi-diurnal 
arc.  The  value  of  the  hour  angle  at  the  time  of 
apparent  conjunction  may  be  determined  by  increas- 
ing that  at  the  time  of  true  conjunction  by  the  qnan 
gd) 

tity ,  according  to  the  tables  on  pages  401 

a,  .J  —  6 

and  402 ;  and  it  may  be  observed  that  this  hour 
angle  must  not  exceed  the  semi-diurnal  arc  by  more 
than  half  an  hour.  For  the  latitude  of  Greenwich, 
the  semi-diurnal  arcs,  allowing  33'  for  refraction  in 
the  horizon,  are  shown  in  the  annexed  table. 

As  a  final  test  for  the  exclusion  of  unnecessary 
stars,  it  is  useful  to  calculate  the  extreme  limits  of 
latitude  between  which  the  star  will  be  visibly  oc- 
culted on  the  earth.  These  will  evidently  appertain 
to  the  extreme  northern  and  southern  points  of  the 
northern  and  southern  limits  of  contact,  determined  as  for  a  solar  eclipse,  a  point 
in  the  northern  or  southern  limit  will  depend  on  the  formula}  Nos.  27,  28,  pages 
359-60.  Thus, 


Dec. 
of 
Star. 

Semi-diurnal  Arcs,  for 
the  Latitude  of 
Greenwich. 

Dec.  North. 

Dec.  South. 

0 

h.   m. 

h.  m. 

o 

6    4 

6     4 

6    9+5 

2 

6  i4      * 

5  54    ^ 

3 
4 

6  19      I 
6  24 

549   ; 

5  43    6 

5 

6  29      5 

5  38     5 

6 

6  34 

533     5 

7 

6  39      ^ 

528     5 

8 

6  44 

523     5 

9 

65o      6 

5,8     5 

10 

655       5 

5i3     5 

5 

6 

1  1 

7     o       ^ 

57 

12 

r         6 
7     6 

5     2     : 

i3 

5 
7    " 

456    6 

fi 

5 

i4 

7  17 

4  5l     6 

v5 

23         6 

4  45     5 

16 

17 

7  28       « 

7  34 

4  4o    A 

434  ; 

18 

7  4o 

4  28 

'9 

747       7 

4    22 

20 

7  53 

4    l5       I 

21 

8     o       7 

4    9    ! 

22 

8     6       6 

4    2 

23 

8  i3       7 

356    6 

24 

8  21 

3  49    ' 

25 

8  28 

34i     8 

26 

8  36 

334    ' 

27 

8  44 

3  26    ' 

28 

8  53      9 

3   18     J 

29 

9     »       ' 

3    9    9 

3o 

9  »V 

3o-9 

cos  w  = 


n±  A' 


M=—  *  ± 


and  thence, 


sin  /  =  sin  D'  cos  Z  +  cos  D'  sin  Z  cos  M. 


20 


402  SPHEEICAL    ASTRONOMY. 

It  is  now  our  object  to  ascertain  what  value  of  «'  will  render  the  value  of  /,  so 
deduced,  a  maximum  or  a  minimum,  and  what  will  be  the  corresponding  value  of  I. 
Let  0  be  an  arc  determined  by  the  equation, 

cos  Z  =  cos  0  sin  w     ..........     (6) 

Then  by  uniting  with  it  the  equation 

cos  <•>'  sin  Z  =  cos  w   .     .     .  .  .    .     (7) 

we  infer  that 

sin  «'  sin  Z  =  sin  <f>  sin  w      .          .....     .     (8) 

because  the  squares  of  these  three  equations  added  together  will  give  unity  on  each 
side.  By  these  equations  we  shall  hence  have 

sin  D'  cos  Z  =  sin  D'  cos  0  sin  w, 

sin  Z  cos  M'=  sin  Z  (cos  i  cos  «'  T  sin  i  sin  w'), 

=  (cos  a/  sin  Z)  cos  t  T  (sin  «'  sin  Z}  sin  i, 

=  cos  i  cos  w  T  sin  t  sin  <f>  sin  w  ; 
and,  consequently, 

sin  I  =  cos  D'  cos  t  cos  w  -}-  sin  w  (sin  D'  cos  0  T  cos  D'  sin  i  sin  0), 
which  now  involves  only  one  variable  <j>.     Again,  assume  two  arcs,  6,  \}/,  which  will 
fulfil  the  equations, 

cos  6  cos  i//  =  sin  J)'       ..........     (9) 

cos  6  sin  i//  =  ±  cos  D'  sin  i  ........  (10) 

A.  third  equation  will  follow  from  these,  viz.  : 

sin  6  =  cos  D'  cos  *  ...........  (11) 

because,  as  before,  the  squares  of  these  three  equations  will  together  make  unity. 
The  value  of  sin  /  will  now  become 

sin  I  =  cos  w  sin  9  -{-  sin  w  cos  0  cos  (0  -f"  t//). 

The  angle  0  +  \f/  being  the  only  variable  in  this  expression,  it  is  evident  that  the 
greatest  value  of  I  will  have  0  -|-  \f/  =  0,  and  the  least  0  -f-  \f/  =  180°.  Therefore, 


These  would  be  the  extreme  latitudes  for  the  appearance  of  the  occultation  if  tl  e 
earth  were  a  transparent  body  ;  as  this,  however,  is  not  the  case,  it  will  be  nect-s- 
sary  that  the  star  should  be  above  the  horizon,  a  condition  not  included  in  the 
preceding  equations.  The  zenith  distance  Z  must  not  exceed  90°,  and  therefore 
cos  Z  must  necessarily  be  a  positive  quantity. 

By  the  equation  (6)  cos  Z  must  have  the  same  sign  as  cos  0,  and  this  must  be 
the  same  as  -|-  cos  i//  for  northern  limit,  or  —  cos  \l  for  southern  limit,  because  in 
'the  former  case  0  -j-  t//  =  0,  and  in  the  latter  ^  -j-  !//  =  180°.  But,  by  (9),  cos  </. 
must  have  the  same  sign  as  JDf.  Consequently, 

For  I  Tuthe'rn  }  limit'  CO8  Z  has  the  8ame  si^n  as  |  -  D'. 

It  is  evident,  therefore,  that  the  extreme  northern  limit  will  have  the  star  below 
the  horizon,  and  be  excluded  when  J)'  is  negative,  and  that  for  the  same  reason 
the  southern  limit  will  be  excluded  when  D'  is  positive.  Thus  the  only  admissi- 
ble extreme  limit  will  be  determined  by  the  equations 


using  upper  signs  when  D'  is  positive,  and  under  signs  when  D'  's  negative. 
The  other  limit  for  the  actual  appearance  of  the  occultation  wil   evidently 


APPENDIX    XI.  403 

of  the  two  places  where  the  other  limiting  line  meets  the  rising  and  setting  limits, 
and  will  bo  determined  by 

C08W  =  —  —  —  ,         sin  J2  =  eos  D'  cos  [  (—  i)  =p  w]  .     .    .     .     (13) 

using,  as  before,  upper  signs  when  D'  is  positive,  and  under  signs  when  D'  is  neg- 
ative. 

The  equations  (11),  (12),  (13),  for  convenience  in  determining  the  species  of  the 
angles,  may  be  put  in  the  following  form  : 


COS  Wi  =  -  -—  -  ,  COS  W2  =       —p  -     j 

sin0  =  cos  D'  cost  V   .     •    •     •     (14) 

I,  =  wi  —  0 
sin  /a  =  T  cos  D'  cos  (w2  —  i)  J 

observing  that  Wi,  Wa,  0,  and  «,  must  here  take  the  same  sign  as  D'  ;  also, 


These  formulae  are  applicable  to  a  solar  eclipse.  For  an  occultation  of  ft 
star  by  the  moon,  Pr  will  be  the  moon's  horizontal  parallax,  and  A'  her  semi- 
diameter,  which,  as  these  limits  are  not  wanted  very  accurately,  may  be  regard'-' 
ed  as  true  quantities  ;  also,  we  may  neglect  u  and  so  take  6  instead  of  D'.  Since 

-^  =  [9.43537]  =  .2725,  the  formulae  for  an  occultation  will  hence  be 
tan  t  = -— -  n  =  (diff.  dec.)  cos  t 

a,  COS  o 


cos  Wi  =  =p  —  —  .2725,          cos  wa  =  T  — -  +-2725 


-   ( 
sin  6  =  cos  5  cos  t 

sin  J2  =  T  cos  6  cos  (w2  — «) 

in  which  we  also  give  to  the  angles  Wi,  w2, «,  9,  the  same  sign  as  6,  and  use  upper 
signs  when  6  is  positive,  and  under  signs  when  S  is  negative.  We  may  also  ob- 
serve, that, 

1.  When  6  is  north,  l\  is  the  most  northern  limit ;  and  when  5  is  south,  l\  is  the 
most  southern  limit. 

2.  When  W!  is  imaginary,  h  will  be  90°,  and  of  the   same  name  as  t.     In  this 
case  the  occultation  will  be  visible  about  the  pole  of  the  earth,  which  is  presented 
to  the  star  ;  the  visibility  will  extend  beyond  the  extremity  of  the  disk  of  the  earth 
as  it  would  be  seen  from  the  star. 

8.  When  w2  is  imaginary,  h  will  be  the  complement  of  6,  and  of  a  different  name 
from  i.  In  this  case,  if  we  consider  the  disk  of  the  earth  as  seen  from  the  i-tar,  the 
visibility  of  the  ocuultation  will  extend  beyond  that  extremity  of  the  disk  whicjh 
has  the  pole  on  the  other  side  of  it. 

After  an  occultation  is  computed  for  any  particular  place,  if  we  deduct  the  star's 
right  ascension  from  the  sidereal  times  of  immersion  and  emersion  we  shall  get 
the  hour  angles  of  the  star,  -\-  West,  —  East.  By  comparing  these  hour  angles 
with  the  semi-diurnal  arc  of  the  star,  we  can  distinctly  ascertain  the  positions  of 
t.he  star  with  respect  to  the  horizon. 


404:  SPHERIC  AL    ASTRONOMY. 


V. — ECLIPSES  OF  THE  MOON  BY  THE  EARTH'S  SHADOW. 

These  may  be  also  resolved  in  the  same  way  as  those  of  the  sun.  The  absolute 
positions  of  the  moon  and  shadow  being  independent  of  the  position  of  the  specta- 
tor on  the  earth,  the  determination  of  parallaxes  -will  be  here  unnecessary,  which 
much  simplifies  the  calculation  of  these  eclipses.  The  considerations  requisite  to 
be  attended  to,  by  way  of  distinction,  are  the  following : 

Semi-diameter  of  the  shadow      =  —  (P1  -J-  TT  —  »). 

oO 

Semi-diameter  of  the  penumbra  =  —  (P1  -f  T  —  o)  -}-  2  ff. 

60 

Right  ascension  of  centre  of  shadow  =  that  of  the  sun  ±  12h. 
Declination  of  centre  of  shadow  =:  that  of  the  sun  with  a  contrary  name. 

The  figure  of  the  eaith  being  spheroidal,  that  of  the  shadow  will  deviate  a  little 
from  a  circle,  so  that,  to  have  a  mean  radius,  the  horizontal  parallax  of  the  moon 
must  be  reduced  to  a  mean  latitude  of  45°.  This  will  give 

P'  =  [9.99929]  P: 
P  denoting  the  moon's  equatorial  horizontal  parallax. 

Also, 

a  =  right  ascens.  moon  minus  right  ascens.  centre  of  shadow  ; 
x  =  (dec.  moon  -f-  a  corr.)  minus  dec.  centre  of  shadow ; 

y  =  a  cos  D. 

With  these  we  compute  according  to  the  equations  (16)  and  (18),  pages  387  and 
368,  observing  the  following  values  of  A'  : 

For  I  h^rnal  \  contact  with  snadow,  A'  =  semi-diam.  shadow  ±  *. 
For  ]  fn^ernal  [  contact  witn  penumbra,  A'  =  semi-diam.  penumbra  ±  «. 

The  angular  positions  of  the  points  where  the  contacts  take  place  will  be  esti- 
mated on  the  circumference  of  the  shadow  or  penumbra  the  same  as  they  were 
before  on  the  limb  of  the  sun.  These  angles  will  therefore  be  in  a  reversed  posi- 
tion on  the  disk  of  the  moon,  and  consequently  as  they  come  out  from  the  compu- 
tation will  have  reference  in  the  first  instance  to  the  inverted  appearance  of  the 
phatse. 

The  relative  orbit  of  the  moon,  not  being  affected  with  parallax,  will  not  sensibly 
deviate  from  a  great  circle  in  the  course  of  the  eclipse  ;  and  hence  the  assumption 
of  the  particular  time,  on  which  to  found  the  calculation,  will  be  but  of  little  im- 
portance: any  convenient  time  may  be  assumed  near  the  time  of  opposition. 

Ct  will  be  unnecessary  to  add  any  further  remarks.  We  shall  conclude  this  pa- 
per with  a  tabular  recapitulation  of  the  formulae  which  relate  to  the  phenomena 
for  a  particular  place,  in  which  eclipses  of  the  moon,  for  the  sake  of  clearness,  are 
given  separately.  The  object  of  this  taMe,  like  the  former  one  for  the  general 
eclipse,  is  to  simplify  arid  expedite,  by  an  -asy  reference,  the  actual  operations  of 
the  computer. 


APPENDIX    XI. 


405 


I.    ECLIPSE   OF   THE  SUN   FOR   A   PARTICULAR   PLACE. 
1.  h  =  apparent  time  of  true  d  m  R-  A.  to  nearest  minute, 

"With  this  as  an  argument,  take  out  the  numbers  fi,  6([\  from  the  following 
table : 


Table  for  reducing  the  true  to  the  app.  (j  in  R.  A. 

« 

I(i1 

6 

*U) 

Hour  Angle 

Hour  Angle 

h 

same 

h 

same 

at  true  <5- 

H  

sign 

at  true  c5. 

+  — 

sign 

as  h. 

as  /*. 

h.  in. 

h.  in. 

h.  m. 

b.  m. 

0   0 

12   0 

25 

0 

3  o 

9  ° 

(8 

7' 

JO 

ii  5o 

25 

4 

10 

8  5o 

»7 

74 

20 

4o 

25 

9 

20 

4o 

16 

77 

3o 

3o 

25 

i3 

3o 

3o 

i5 

79 

4o 

20 

25 

'7 

4o 

20 

i4 

82 

5o 

10 

25 

22 

5o 

10 

i4 

84 

I   0 

II   0 

24 

26 

4  o 

8  o 

i3 

87 

10 

10  5o 

24 

3o 

IO 

7  5o 

12 

89 

20 

4o 

24 

34 

20 

4o 

II 

9l 

3o 

3o 

23 

38 

3o 

3o 

IO 

92 

4o 

20 

23 

42 

4o 

20 

9 

94 

5o 

10 

22 

46 

5o 

IO 

8 

95 

2   0 

IO   0 

22 

5o 

5  o 

7  o 

7 

97 

IO 

9  5o 

21 

54 

IO 

6  5o 

5 

98 

2O 

4o 

21 

57 

20 

4o 

4 

98 

3o 

3o 

2O 

61 

3o 

3o 

3 

99 

4o 

20 

'9 

64 

4o 

20 

2 

100 

5o 

IO 

'9 

68 

5o 

IO 

I 

IOO 

3  o 

9  o 

id 

7* 

6  o 

6  o 

O 

100 

Then.  T  denoting  the  approximate  mean  time  of  app.  (5,  in  units  of  an  hour, 

|(H 

T—  mean  time  true  (5  -\ , 

a,./ — 6 

in  which  ai  must  be  used  in  minutes  of  are ;  also/=  -— ^,  is  a  factor  depend 

cos  / 

ing  on  the  latitude,  which,  for  several  .principal  observatories,  is,  for  conTenienc* 
included  in  the  following  table : 


406 


SPHERICAL    ASTRONOMY. 


Auxiliary  Quantities  depending  on  Geographical  Position. 

Place. 

P 

cot  I 

cos  I 

/ 

Longitude. 

Aberdeen 

9.99900 

+  9.81289 

9-73637 

3.12 

h.  m.    s. 
W.  o    8  23 

Altona    .... 

9.99908 

+  9.87133 

9.77576 

2-85 

E.   o  39  47 

Berlin     .... 

9.99910 

+  9.88751 

9-78603 

2.79 

E.   o  53  36 

Bedford  .... 

9.99911 

+  9.89345 

9-78974 

2.76 

W.  o     i  52 

Cambridge   . 

9.99911 

+  9.89231 

9-78903 

2-77 

E.   o     o  24 

Cape  of  Good  Hope 

9.99966 

—0-17494 

9.91980 

2»o5 

E.    i   i3  55 

Dublin    .... 

9.99909 

+  9.87385 

9.77737 

2.84 

W.  0    25    22 

Edinburgh   . 

9.99902 

+  9-83256 

9-75ooi 

3-o3 

W.  o  12  44 

Greenwich   . 

9.99913 

+  9-90381 

9.79610 

2*72 

o     o    o 

Ormskirk 

9.99908 

+  9-87092 

9.77549 

2-85 

W.  o  ii  36 

Oxford    .... 

9.99912 

+  9-89939 

9.79340 

2.  74 

W.  o     5     2 

Kensington  . 

9.99918 

+  9-90340 

9.79586 

2.72 

W.  o    o  47 

Milan      .... 

9.99928 

+  9.99577 

9-84736 

2.42 

E.    o  36  47 

Paris       .... 

9.99920 

+  9-94451 

9-81997 

2.58 

E.   o    9  22 

Slough    ...      . 

9.99913 

+  9.90337 

9-79584 

2.72 

W.   0       2    24 

2.  The  time  T  being  computed  to  the  nearest  minute,  take  out  the  correspond- 
ing values  of  P,  ir,  <r,  5,  from  the  Ephemeris ;  and  prepare  the  constants 

c  =  [4.68555]  p, 

A  =  c(P  —  n),  m  =  A  cos  I, 

Q,  =  [4.7172],  Qi  =  mQt  sin  i, 

s  =  [9.43537]  P. 

3.  Take  out  D,  t,  a,  J)lf  ait  for  the  time  T. 

h  =  sidereal  time  at  place  minus  J>  's  right  ascension,  to  the  tenth  of  a  minute, 
In  are. 


K  zzzz 


n  =  k  cos  A, 


A  a  =  [5.31439]  k  sin  A  [corr.  for  M], 

A  01  =  Qi  n,  A  Di  =  Q2  sin  A. 

Correction  for  n  to  be  taken  from  the  table  on  page  383. 

(A)  =  A  -f  i  A  a, 

tan  6  =  cos  (A)  cot  /,  G  =  cos  (A)  cos  /, 

sin  " 


tan  M  i^. 


cos  (e  -f-  Z>) 


tan  (A),  tan  e  =  tan  (6  -f-  D)  cos  Jff 

B  =  cos  Jf  cos  t ; 

sin  S  # 


J/  to  be  in  the  same  semicircle  with  A. 


APPENDIX    XI.  4-07 

M,  =  A  sin  e,  A  D  =  [5  31439]  A  B  [corr.  for  «i], 

s'  =  s  [corr.  for  ni\. 

Fo(          partial  J    ^   A,=   U  +  .) 

(  total  or  annular  )  (  «'  ~  o  ) 

Correction  for  Wj  to  be  taken  from  the  table  on  page  383. 

D'  =  D  —  A  D,  a'  =  a  —  A  a, 

y  =  (a  —  A  a)  COS  Z>',  y:  =  (ai  —  A  «i)  COS  J9', 

«  =  (2)'  -f-  a'  corr.)  —  S,  x,  =  A  —  A  Di. 


sin  S       cos  /S' 

TFcos*  [8.55630 
TP  cos  [  —  (S  -f  «)],  ^  =  --  -  - 


n  IT 

7.  COS  W  =  -  :,  C  = 

A  '  cos  ( 


=  e  sin  a,  <2  =  c  sin  b. 


ending     ) 
Time  of  greatest  phase  =  \  sum  of  times  of  beginning  and  ending. 

When  n  <  «'  ~  »,  the  eclipse  will  be  total  if  s'  >  »,  or  annular  if  «'  <  v.  in  thia 
ease  these  last  equations  No.  7  must  be  repeated  for  this  phase  with  A'  =  d  ~  * 
the  results  of  which  ought  to  give  the  same  time  for  the  greatest  phase. 

Take  A  '  for  partial  phase,  and 

Portion  of  sun's  disk  eclipsed  =  A  '  —  n. 

Magnitude  of  eclipse  =  —      —  ,  the  sun's  diameter  being  unity. 

8.  For  the  positions  of  the  points  of  contact  on  the  limb  of  the  sun, 
At  |  be^^  |  ,  angle  from  north  towards  east  =  |  <(  ~  |  j  ~  ^  for  direct  image. 

At  \  be^'"ning  1  ,  angle  from  north  towards  east  =  j  («0»  -  •)  -  .  )  for  inverted 
i     ending     )  (  (180°  —  1)+«)       image, 

For  the  position  of  the  moon's  centre  at  greatest  phase, 

Angle  from  |  "^^  I  towards  east  =  ||  "~  ||  I  for  direct  image. 

Angle  from  |  ^  J  toward,  east  =  |  ||^  -  0_  ^  |  for  inverted  [m^ 


408  SrrfERICAL    ASTRONOMY. 

9.  For  a  more  accurate  calculation  of  the  time,  <fec.,  of  beginning  of  the  partial 
phase,  assume  a  convenient  time  near  to  the  preceding  determina-tion.     For  this 
time,  take  out  the  quantities  D,  Dlt  t,  a,  ah  from  the  Ephemeris ;  and  proceed  as 
in  Nos.  8,  4,  5,  6,  7,  omitting  b,  t^  and  the  times  of  greatest  phase  and  ending. 

Let  Mi,  «i,  «»i,  be  the  values  of  the  angles  in  this  computation ;  then,  for  the  po- 
sition of  the  point  of  contact  on  the  limb  of  the  sun, 

Angle  from  •}  '    t  towards  the  east  =  ?  } '  ~~'1'       Wl  (  for  direct  image. 

(  vertex  )  (  (  -_i,)_Wl_  Mi  ) 

Angle  from  <(  north    \  towards  the  east  =  \  0  80°  ~  ")  ~  "'  \  for  inverted 

(  vertex  J  (  (180°  —  ix)  —  Wl  —  Ml  )       image. 

10.  For  a  more  accurate  calculation  of  the  time,  <fec.,  of  ending  of  the  partial 
phase,  assume  a  convenient  time  near  to  the  first  determination.     For  this  time, 
take  out  the  values  of  D,  l)lt  i,  a,  aj;  and  proceed  as  in 'Nos.  3,  4,5,  6.  7,  omitting 
a,  ti.  and  the  times  of  beginning  and  greatest  phase. 

Let  Jf2,  io,  «2,  be  the  angles  in  this  computation ;   then,  for  the  position  of  the 
point  of  contact  on  the  limb  of  the  sun, 


Angle  from  \  north    I  towards  the  east  =  j  (       ">)  +  <**  I  for  <#mi  image. 

(  vertex  )  {  (  —  ,a)  +  U2  —  Jfa  ) 

Angle  from  J  north    i  towards  the  east  =  |  (18°"  ~  ">  +  W2  I  for  inverted 

(  vertex  )  ((180°  —  «2)-f-<«>2  —  Jfa  J 


image. 


II.  -  FOKMUL^E    FOB   REDUCTION    TO   DIFFERENT   PLACES. 

11    Instead  of  Nos.  5,  6,  7,  substitute  the  following: 

D'  =  D  —  A  D,  «'  =  a  —  A  a, 

a?,  =  Z>!  —  A  A,  y!  =  (a,  —  A  o!)  cos  D', 

tan  «  =  -,  k  =  [3.55630]  A'cos' 


'  corr.)  —  i  a  cos  D' 

y  cos  ^  =      -  -  --y—  ^  -  ,  y  sm  ^  =  -  -^—  , 

=  y  cos  (^  -f  «),  ?  =  ^X  sin  (<£  +  i), 


[5.31439]  A  r 
12.  6  =  =  -  r-J  —  fcorr.  for  nil 

A 

e  in  minutes  =  [7.9208]  A  a  sin  D,  %  =  (90°  +  «)  —  #. 

18.  ^Tr=  the  true  Greenwich  hour  angle  of  J)  at  the  time  T. 

jj  jj< 

-r-  =  COS  D  COS  I.  7T  =  COS  D  sin  t 

0  to 

rcos  (0/  —  ^)  =  sin  D  cos  «,  -^r  cos  fy"  —  -ff)  =  sin  D  sin  », 

c« 

-  wn  (^'  —  £T)  =  cos  x,  rr  sin  <^"  —  //)  =  »in  X- 


APPENDIX    XL  400 

14.  The  constants  T  ',  k,  p,  IS,  L",  y',  y",  being  so  computed,  the  angle  u  and 
the  time  t  of  the  phase  for  any  place  whose  north  latitude  is  /  and  east  longitude 
X,  will  be  determined  by  the  two  following  equations,  in  which  the  upper  sign  re 
lates  to  the  beginning  and  the  under  sign  to  the  ending. 

cos  w  =  p  —  L'  sin  I  -f-  y'  cos  I  cos  (A  -f-  </•'); 

t  =  T  T  k  sin  w  -f-  L'1  sin  I  —  y"  cos  I  cos  (A  -f  ^''). 

The  result  will  be  the  most  accurate  when  the  place  is  near  to  that  on  which 
the  previous  part  of  the  calculation  is  founded. 

in.  -  TRANSIT  OF  MERCURY  OR  VENUS  OVER  THE  DISK  OF  THE  SUN. 
(Same  notation  for  the  planet  as  for  the  moon.) 

15.  Assume  the  time  Tnear  to  the  time  of  conjunction  in  longitude,  or  right  a» 
cension. 

a  =  sun's  right  ascension  —  planet's  right  ascension  in  arc; 
aj  =  hourly  variation  of  a  ; 
J)1  =  sun's  hourly  motion  in  declination  minus  that  of  the  planet. 

5> 
*. 

For  contact  of  planet's  centre  with  sun's  limb,  A  =  «•* 

A 


For  |  exter?or  I  contact  of  limbs,  A  =  \ff  + 
(  interior  )  (  a  — 


tan  t 


«!  cos  & 
cos  t 


cos  <5  A 

(6  -f-  a  corr.)  —  D  a  cos  £ 

y  COS  t//  = ' ,  y  sin  i//  =  , 

A  A 

cos  w  =  y  cos  (t/>  -f-  «)>  y  =  A;  y  sin  (<//  -f- 1). 

16  JEf=  the  true  Greenwich  hour  angle  of  O  at  the  time  2T 

sin  w  ' 
-jji  =  cos  i  cos  [(  —  «)  T  w] ; 

—  cos  (^"  —  #)  3=  sin  8  cos  [(  —  i)  T  w] ; 
£77  sin  (^"  —  ZT )  =  sin  [( —  i)  T  w]. 

17.  Then,  for  the  centre  of  the  earth, 

and,  for  any  place  whose  latitude  is  /  and  east  longitude  A, 

t  =r  (/)  qp  [y"  f  cos  J  cos  (A  -f  »//")  —  Z"  />  sin  l\ 
osinp'  the  uppe-  signs  for  the  ingress,  and  the  under  signs  for  the  egress. 

The  positions  of  the  points  of  ingress  and  egre<sr  estimated  from  the  north  point 
of  the  sun's  limb  towards  the  east,  as  the  transit  would  be  seen  from  UK  centre 
of  the  earth,  will  be  determined  in  the  same  manner  as  for  the  immersion  and 


±10  SPHERICAL  ASTRONOMY. 

emersion  of  an  occultation,  No.  19,  using  w  for  &>.  These  angles  may  be  assumed 
to  be  the  same  for  any  place  on  the  surface,  the  effect  of  parallax  being  so  very 
minute. 


IV.  -  OCCULTATION   OF   A   STAR   BY   THE   MOON. 

GENERAL  LIMITS  OF  LATITUDE. 
1&  (ai  and  Di  at  true  6  )• 

tan  i  =  •  —  —  —  ,  n  =  (diff.  dec.)  cos  «. 

«i  cos  6 

cos  Wj  =  =F  -p  —  .2725,  cos  w2  =  =F  —  -f  .2725, 

sin  0  =  cos  i  cos  t, 

li  =  Wi  —  Q,  sin  /2  =  =p  cos  &  cos  (wa  —  «)> 

w^  Ws,  i,  0,  same  sign  as  6, 

v-  555S 

When  Wi  is  impossible,  l\  =  90°,  with  the  same  name  as  3. 

When  Wa  is  impossible,  /a  =  complement  of  <5,  with  different  name  from  i. 

CALCULATION  FOR  PARTICULAR  PLACE. 

19.  For  the  latitude  of  the  place  prepare  the  constants 

dd) 
?(')  =  p  cos  /,  $  :»  =  f  sin  /  =  -,  f  (»)  =  [9.41916]  *(»>, 


rhich  will  serve  for  all  occultations  at  that  place. 
For  the  time  of  true  <5  find 

h  =  sidereal  time  at  place  —  right  ascension  of  star  ; 

and  thence  determine  the  time  T,  as  in  No.  1.     For  this  time  take  out  the  quanti- 
ties P,  s,  D,  Di,  a,  ai  ;  and  compute 

x  =  (D  —  6)  —  ($W  .  P  cos  a  —  f'V  .  P  sin  6  cos  h)  ; 
y  =s  a  cos  3  —  0'1)  P  sin  A  ; 
^  =  D!  —  #(3)  .  P  sin  5  sin  /<  ; 

^!  =S  Oj  COS  ^  -  0(3)  .   P  COS  A. 

With  these  proceed  as  in  Nos.  6  and  7,  using  A  '  =  s  =  [9.48587]  P. 

20.  For  the  positions  of  the  points  of  immersion  and  emersion  on  the  limb  of  the 
moon, 

At  (  immersion  j  |  ,  angle  from  north  toward8  east=  j  (180°-«)-«  )  for  ^       . 
<   emersion    )'  (  (l80°-«)  +  w  ) 

At  \  Immer810n  j.   angle  from  north  towards  east  =  •!  (—')-«  I  for  inv«r^  image. 
(  emersion   )  (  (  —  «)  +  <"  ) 

For  the  same  angles  from  the  vertex  we  must  deduct  the  parallactic  angle  for 
each  time. 

21.  If  an  accurate  calculation  is  wanted,  proceed  as  with  a  solar  eclipse. 


APPENDIX   XI.  ill 


V.  -  ECLIPSE  OF  THE  MOON. 

i2.  Fix  on  a  convenient  time  near  to  the  time  of  opposition  in  longitude,  or  full 
oon  ;  and  for  this  time  find  P,  s,  «•,  <r, 

a  =  D's  right  ascension  minus  (©'s  right  ascension  ±  12h),  in  arc; 

ax  =  hourly  motion  of  a  ; 

x  =  (  D  's  dec.  +  a  corr.)  plus  O's  dec.  ; 

Xi  =  hourly  motion  of  x  ; 

y  =  a  cos  D  's  dec.  ; 

yl  =  ai  cos  ])  's  dec. 

P1  =  [9.99929]  P. 


23'  Semid.  shadow      = 


Semid.  penumbra  =  —  (P'  +  *  —  e)  +  2  c. 


For  I  fnternia  [  contacfc  with  shadow»  A/  =  semid.  shadow  j  it  f  *• 
For  •]  pxternal  L  contact  with  penumbra,  A'  =  semid.  penumbra  -J  _  f  «• 
The  remaining  computation  as  in  Nos.  6  and  7. 

24.  For  the  positions  of  the  points  of  contact  on  the  limb  of  the  moon, 

At  \  immersion  {.  ,  angle  from  K  towards  R  =  \  (180°0  ~  '>  ~  w  I  for  direct  image. 
<  emersion     )  (  (180   —  »)  -f  w  ) 

At  \  iramersion  I  ,  angle  from  N.  towards  E.  =  \  ^  l\  ~  w  [  for  inverted  image. 
(  emersion     )  ((  —  i)  -f  w  J 

At  the  middle  of  the  eclipse, 


/  cent,  shadow  from  N.  towards  E.  =J  (180°  ~  f)  [  for  ]  dim*    ,  J  image. 

(  (—  t)  )        (  inverted  } 

To  get  the  same  angles  from  the  vertex,  the  parallactic  angle  must  be  deducted 
for  the  respective  times. 


J3  &  5*-O 


412  SPHERICAL    ASTRONOMY. 

Examples. 

I.  -  ECLIPSE  OF  THE  SUN. 

Let  it  be  required  to  calculate  the  circumstances  of  the  solar  eclipse  of  May  1ft 
183G,  as  it  will  be  seen  at  the  observatory  of  Edinburgh. 
The  elements  of  this  eclipse  are  stated  at  page  362. 

h.    m.    s. 
Greenwich  sidereal  time  at  Greenwich  ) 

mean  noon 
Longitude      ........          12  43-6  W. 

Edinburgh  sidereal  time  at  Greenwich  )  „ 

mean  noon       .......  [3  2O  '*4     "      '     *~  '? 

Sun's  right  ascension  at  d    .      .      .      .     3  29  25-2     f     .       3  «o3 

Hour  angle  h  at  Greenwich  mean  noon  —  o     910-8  83  •  i 

(  Greenwich  mean  time  of  6          ..     2  21   22-9  -8 

I  Acceleration   .......  23  •  2       aj  ./  84  .  — 

Aatc5        ...      +  2~73     '.     .      6+21     SO  +  55   (4-  -87 
a,./_g+    63  5o-4 

T6 
h.  m. 

Greenwich  mean  time  of  true  c5     .       2  21 
g<i)_i-(ai.y_fi)    .      .      .    +     52 

3~r3 


-  9-99902 

.  4-68555 

.  T6845^ 

.  3-5i254 

.  8-19711 
.'  9-75001 

.  7.94712 
.  4-7172 
+  9-5i2i 

Q2  +  2-1764 

COMPUTATION  FOR  3h  i3m,  Greenwich  time, 
o     '     "  o     '      "  '     " 

D  +  19  33  43  a  +  18  58  29  a  +  23  49 

Di  +         9  19  ai  +  27  43 

h.    m.     s. 
Edinburgh  sidereal  time  at  Greenwich  mean  noon       .      .      .      •     3  20  i4'4 

(  3h  om 3    o  29-6 

Sidereal  equivalent  for  -Jo  3       . 

6~33  46-i 
Moon's  right  ascension 3  3i     9-0 


P   . 

If 
P  —  v      . 

logP       . 
const. 

log  a 
o  . 
i     + 

T     . 
CONSTANTS. 

54  23-4 
8-5 

const. 
c 

A      . 
cos  / 

m 
sin  & 

54  i4-9 

3-5i367 
9-43537 

2.94904 

1  5'  49"  -9 

18°  58-  5     .      . 

{time      .      .      .     3     2  37-i 
arc  ....  +T5T3o~73 


APPENDIX    XI. 


m     .      .     7-94712 
cos  D     ,      .     9.97418         const. 

5.3r439 
7.97294 

cos  h      .       +  9.  84446         sin  h      .       + 
n      .       +  7-81740         corr.  for  n   . 
Ql    .      .     4-7I72        j  log   .      .       + 

9-8544o    .     .     . 
286 
3-i4459        Q,     .     . 

+  9-8544 
+  2-1764 

{log    .      .       +  2-5346        I  A  a       .       + 

23'  1  5"       j  log   .     . 

+  2-o3o8 

A*!              -             +      5'    42" 
0           / 

h  .     .     +45  39-3 

£  A  a         +          II  .6 

1  A  A      . 

+  i'  47" 

(h)      .     +  45  5o-9    .    cos  .     .     .     + 
cot  I      .      .      + 

O        ' 

9.84296  .... 

9-83256        cos^     . 

+  9-84296 
,      +  9-75001 

0  +  25  20-9         tan  9      .      .      + 

9-67552             G    . 

.      +  9-59297 

D  +  19  33-7         sin  6       .      .      + 
9  +  D  +  44  54-6    .    cos  .     .     .     + 

9-63i56 
9-85017             B    . 

.      +  9.81158 

+ 
tan  (h)   .      .      + 

9-78139    .    check  . 
0-01286 

+  9-78139 

0     ,          (  tan  M    .      .      + 
*,r  i    o  ,    K  /   K. 

9-79425 

M  -f-  o  I    D4  •  3      "S 

(  cos  M    .      .      + 

Q-Q2885   . 

4-  0-02885 

tan  (6  +  D)       + 

9-99864        cos«    . 

.      +  9-88273 

o      ,           (tan£       .      .      + 
«  +  4o  i4-3     J  cos  £      .     .     + 

9-92749             B  . 
9-88273         const.  . 

.      +9-8n58 
.      +  5-3i439 

(  sin  e        .      .      + 
A     .     . 

8  •  i  97  i  i 

5-12597 
8  •  1  97  1  1 

Wi        .        .        + 

8-00733         corr.  for  n 

h               444 

(log       - 

3.32752 

log  «... 

2.94904      1   A  D   . 

.     +  35'  26" 

444 

(  log   .      .     .     . 

2.95348 

.  :  :  :  : 

i4'58".4           A. 
i5  49  -9      A  A  . 

.      f  9  19 

.     +  i  47 

Partiil  A'   .      .      .      . 
Annukr  A'       ... 

3o  48  -3           *,   . 
o  5i   .5 

.      +7  32 

0         "          ' 

Z>  ...     4-  19  33  43                        a  + 

A  D   .      .       f         35   26                         L  a  + 

i     it 
23  49        .                <»t 

23    15                           A  a, 

-f  27  43 
+    5  42 

/>'...      +  18  58  17    j                     a'  + 
a'  corr.      .                       o    |  log              +  i  • 
S    .     .      .      4-  16*  58  29      cos  D'         +  9- 

o  34      ( 

53i48     \  log       .     + 
97574    .     .     .     .     + 

+   22       I 

3  .  i  2090 
9.97574 

—  O    12 


1.50722       yi 


+  3-09664  (I) 


SPHERICAL    ASTRONOMY. 


0         '                / 

8  .     .    +  no  28-0  .  -j 
i  .      .    +     19  53-5 

y   -    • 
*   . 

tan£    . 

cos  S    . 
W  .     . 
cos  . 

ft 

log  A'  . 

COS  W       . 

+  1.50722 
—  I  -07918 

yi    .    +  3-09664  (i, 
a?»     .    +  2-655i4 

—  0*42804 

cot  i    +  o-44»5o 

—  9.54364 

cos  t    +  9-97328 
.     .      +  1-53554 
const.       3-5563o 

+  5^>65iT  (2) 

+  1-53554 
—  9.81129 

—  (S  +  «)  —  i3o  21-5    . 

Partial   .    . 
w    +    90  4i«i 

—  1-34683 
3-26677 

//+  i-96848  (2)  —  (i) 
.      .     —  8-08006 

—  8  -  08006 

a  —  221     2-6               c     . 
b   —    89  4o-4         sin  a 

ti 
Assumed  time. 

Beginning  .... 
Longitude  .... 
PARTIAL.          Beginning  .... 

n     . 
Annular   .    .  log  A'  . 
»  +  n5°33'-9       cosw     . 
-  (S  +  «)  —  i3o   21  .5              c     . 

—  3-88842 
+  9.81732 

c  —  3-88842 
sin  6  —  9-8o5io 

Greenwich 
mean  times 
V. 
Edinburgh 
mean  times 

Greenwich 
mean  times. 
V. 
Edinburgh 
mean  times. 

—  3-7o574 

+  3-  69352 

h.  m.    8. 

—  i  24  39 
3  i3 

h.  m.   s. 

«,    +     I     22     l8 

.       .       .      3    i3 

i  48  21 
12  44 

Ending  4  35  i8-j 
W.       .      .       12  44  ^ 

i  35  37 

Ending  4  22  34  -j 

—  1-34683 
i  -71181 

H+  1-96848 
—  9-635o2 

—  o-635o2 

—  2.33346 
+  9.9604? 
—  2-29393 

c  —2-33346 
sin  b  —  9-40711 

a  —  245    55  '4       sin  a 
b—    14  47  .6 

tj 
Assumed  time. 
Beginning  .... 
Longitude  . 
ANNULAR.        Beginning  .... 

+   i  -74057 

h.  m.    s. 
—  o     3  17 
3  i3 

h.  m.    s. 
<2  +  o     o  55 
.      .      .     3  i3 

3    9  43 
12  44 

Ending  3  i3  55  j 
W.      .     .      12  44  \ 

2  56  59 

Ending  3     i    n  •] 

POSITIONS  OF  CONTACTS  FOR  DIRECT  IMAGE. 


Partial  contact 


&t  j  beginning 


ending 


Annular  contact  at 


beginning 
ending   . 


(  — 0   —19-9 

*>      +  90-7 


no-6  )  (  west 

g  c  from  north  to  wards  1        . 

o 


n5.6 
i35-5 
95 


:l\ 


(  west 
from  •  orth  towards! 


APPENDIX    XI. 


415 


For  the  same  angles  from  vertex  we  must  estimate  them  towards  the  east,  and 
deduct  the  angle  M,  thus 

o  o 

Beginning  — t35«5  Ending   +  96-7 

M       +    3i-9  M       +  3i-9 

1 67  «4  towards  west.  63-8  towards  east 

COMPUTATION  FOR  i h  48m.  FOR  AN  ACCURATE  DETERMINATION  OF  PARTIAL  BEGINNING. 


D    +  19  19  35-9               $    +  i 
Di   +         9  26 

Edinburgh  Sid.  Time  at  Greenwich 
Sidereal  Equivalent  for  -5 

H  's  R.  A. 

8  67  39-3               a    —  i5  23.2 
ai   +  27  38 

h.  m.    s. 
i  Mean  Noon    .               3  20  i4«4 

.     .            48    7.9 

5    8  32.2 
3  28  18-2 

m     .      .         7-9^712 
cos  D    .         9.97481 

{time 
arc 

const.    . 

+      26°  3'.  f 

5.3i439 
7.97231 
+  9-62690 

+  2.91730 
+  i3'46"-6 

+  9.96666 
+  9-83256 

k     .     . 
cos  h 
n     . 

log  .      . 

h      .     . 
i  A  a      . 

(70  .      . 

e    .    . 

D    .     . 

e+D  . 
Ml        . 

•    •     • 

+  9.96707 
7-92938 
'4-7172 

sin  A     . 
corr.  for  n  . 
(  log  .      .      . 
1    A  a        .       . 

COS  . 

cot  /      .      . 

tan  6     .      . 
sin  0      .      . 
cos  . 

tan  (h)  .      . 
j  tan  Ml 

\  cos  Mi  .      . 
tan  (6  +  D) 

(tan        .      . 
J  cos  .      .      . 

sin  . 
A    ... 
n     . 

#a    .       .       .           2-1764 

j  log  .      .     .    +  i-8o33 
(  A  Di    .     .    +     i'  4" 

+  9.96666 

+      7'  23" 

+  2°5       3-5 

+         6-9 

+  26  10-4 

+  3°i  36.7 
+  19  19-6 

cos  I      .      .    +9.76001 

+  9.78922 

a  .    .    .  +9.70667 

B    .      .      .    +  9.78666 
.      .  check      .    +9.92002 

+  9.71946 
+  9.79945 

+  9-92001 
+  9.67209 

+  5o  56-3 

0            t 

+  21    21  •  I 
0  '    ' 

+  48  55-9 

+  9-69210 

+  0-09068 

cos  e     .      .    +  9.81764 

+  0-06979 
+  8-19711 

B    .     .     .    +9.78666 
const.    .      .          5«3i439 

5«  ioio4 
*     8.19711 
corr.  for  Wi                   5i8 

J-  8-  o7444 

j  log  .      .      .         3-3o333 

.   +33'3o"-6 


416 


D' 

a'c 
^ 

X 


SPHERICAL 

ASTRONOMY 

log*     . 

2.94904 

5i8 

(log        • 

2.95422 

Di 

.     .    4-  9   26" 

iv  .  . 

i5 

o-o 

A  D1 

..4-i4 

a 

i5  4 

:9.9 

x,     . 

,        .      4-    8      22 

A'  .       . 

3o  49-9 

0      • 

n 

t 

a 

t       a 

4. 

1919 

35-9 

a 

— 

i5 

23  '2 

«i 

.     + 

27  38 

+ 

33 

3o-6 

A  « 

+ 

i3 

46-6 

A  «i 

.     + 

7  a3 

jjh 

1846 

5-3 

(        o.' 

— 

29 

9.8 

4- 

20     1  5 

2-2 

(log 

3. 

24299 

log. 

-   +. 

3 

.08458 

i- 

18  57 

39.3 

cos  D'  . 

+ 

9' 

97627 

•      • 

-   + 

9 

.97627 

. 

it 

3i-8 

y   •    • 

— 

3- 

21926 

2/i   • 

.    4- 

3 

•06086  (r) 

x    . 

—  r 

2- 

83998 

xl    . 

.   4- 

a 

.  70070 

O         ' 
-    112  39.8l 

j  tan  S   . 
(sin  S    . 

•f 

0- 

9" 

37928 
96610 

COt  ii 
COS  «! 

•   + 
•    + 

0 

9 

.36~o75~ 
^62^5 

h 

23  34.49 

W.      . 

4- 

3. 

25416 

.    • 

.    + 

3 

.26416 

\- 

89 

5-32 

cos 

4- 

8- 

20168 

const. 

3 

-5563o 

/ 

n    . 

+ 

6 

•77261  (2) 

i  • 

45584 

log  A'. 

+ 

3- 

26716 

H  . 

.   4- 

3 

•  71176  (2)  —  (i) 

f- 

89 

6-93 

COS  0>i    . 

+ 

8^ 

18869 

-     - 

-   + 

8 

•18869 

- 

o 

TTeT 

c     . 

-   + 

5 

•62307 

6-46373 
•  61  —  o-2o683 


— 2-i9363 

tl   .      .    —  oh  2'"363 
Assumed  time        i  48     o 


Beginning 
Long.        . 


I   45   24GrecnhM.T. 

12  44  W. 


32  4oEdin.  M.T 


PARTIAL.  Beginning      . 

If  th^  -jalculation  be  repeated  for  the  Greenwich  time  ih  45rn,  it  will  lead  to  ex 
«tly  the  same  result,  which  is  therefore  to  the  accurate  second,  according  to  the 
data  employed. 

POSITION  OF  CONTACT  FOR  DIRKCT  IMAGE. 

o 
(  — 1,1  .  .    —    23-6 


—  112.7 
4-    21.4 


(  —  tl)  —  wi  —  J/i      .     .    —  i34-i 
The  point  of  contact  is  therefore  j  j.jj°  ]  from  j 


-  towards  west 


APPENDIX   XI. 


417 


II. EQUATIONS    FOR   REDUCTION   OF   PARTIAL   BEGINNING. 

The  data  for  this  computation  are  taken  from  the  preceding  one. 


3.5563o                          5-  3  i  439 
A'         3-26715           A    .        8.19711 
cost       9-96215     corr.  for  MI               5i8 

7-9208 
A  a    .    +2-9173 
sin  D      +9.5198 

6-78560                            3.5i668 

+  0-3579   •     •  e   +     °    2-3 

y,           3.o6o85             A'           3-267r5 

k       +3.72475            6     .    +0-24953 
k     .    +3.72475 

kb.    +3.97428 

0         ' 

D   +  19  19  35.9) 
a  corr.                      2.2) 

90°    +  t    .       n3  34-5 
x    .       n3  32  2 

a     —  2.96530 

cosD'    +9-97627 

t    +  iS  57  39.3 

A'y  sin  i//   —  2-94157 

+        21  58-8     . 

.      .         A'y  cos  \Li   +3-  1  20  1  8 

U,    —33  32-2 

(  tan  i//    —  9-82139 

,      +  23    34-5 

{cos  i//    +  9-92092 

A'y.      .    +  3*19926 
A'       .      .          3.26715 

cos  (d  +  i)    +    9.99341 
j        +    9.92552 

Long.          3  10  .9  W. 

sin  (^  +  i)    —  9«238o2 
k    .      .         3.72475 

j                  —2-  89488 

(  q    .     .    —  oh  i3'n  5s 
T  .     .    +  i    48 

II  +  28  :4  -4 

cos  D    +  9.97481 
cos  i      +  9.96215 
6       .    +0-24953 

L     .    +0-18649 

sin  D     +9.51977 
cos  «      +  9.96215 

T  .     .    +  2     i    5 

cos  D    +  9-97481 
sin  t      +  9-60200 
kb    .    +3-97428 

x"  .  +3.55109 

sin  D   +  9-51977 
sin  i      +  9-60200 

+  9-48192 
cos  x     —  9«6oi34 

+  9-12177 
sin  x     +9-96228 

0      ,       j  tnn    .    —  0.11942 
V  ~~                            /sin     .    —  9.90109  ^ 

(tan   .•    +  o-84o5i 

1  4?"  \,--u  .  +9-9955, 

#+28    i4-4               +9.70025 

tf+   28    14  -4               +  9-96676 

^'     .    —  24    32  «4     6       .         0*24953  ^ 
y'      .    +9-94978 

1         +  TIO     i  -5    kb    .        3.97428 

y"      .     +3.94104 

27 


418  SPHERICAL    ASTRONOMY 

We  have  hence,  for  the  Greenwich  time  t  of  beginning,  at  any  place  whose  lat- 
itude is  I,  =  north,  —  south,  and  longitude  X,  -f  east,  —  west,  the  two  following 
equations,  which  may  be  safely  depended  on  for  any  place  in  Scotland  or  the  North 
of  England. 

cos  w  =0-84240  —  [0-18649]  sin  J+ [9-94978]  cos/ cos (X—   24°3a'-4) 
f  =  2h  im5»  —  [3 .72476]  sin  w  + [3- 55109]  sin/  —  [3. 94104]  cos  /cos(A  +  110°    i'-5) 

Contact  on  0's  limb,  w  -f  23°  34' -5  from  the  north  towards  the  west. 
As  a  check  on  this  calculation  take  the  assumed  radical  place,  Edinburgh,  and 
/  =  + 55°  46'-9,  A=  —  3°  io'-9,  giving  w  =  89°  6'- 9  and  t  =  i1-  45m  24s,  which 
perfectly  coincide  with  the  results  of  the  original  calculation. 

Similar  calculations  for  the  ending  of  the  eclipse  give  the  equations, 

cosw— 0-93848  — [0.20291]  sin^ +[9- 88677]  cos  I  cos  (X+    27°  6'. 7) 
<=ih38m338+[3.6689o]!*inu>+[3.35544]sin/—  [3.90073]  cos /cos  (A  +  1 53°  3' -8) 

Contact  on  0's  limb,  w  —  16°  56' -2  from  the  north  towards  the  east. 
Also  by  calculating  with  T  =  3h  i3'n  for  the  annular  phase  there  will  result 

cos  w  =  2  9. 66600—  [i-75 1 59]  sin/ +[1-46950]  cos /cos  (A  +      i°42'«4) 
f  =  i  h43m78T  [2- 1 4475]  sin  w  + [3-45484]  sin/- [3.9255o]  cos/ cos  (X  +  r3i°55'.9) 

Contact  on  0's  limb,  —  19°  53'- 5  T  u  from  the  north  towards  the  east, 
the  upper  sign  appertaining  to  the  beginning  and  the  under  sign  to  the  ending, 
If  cos  o)  >  i,  the  place  will  be  without  the  Unfits,  and  the  eclipse  "will  not  be  annular. 
By  taking  /  =  +  55°  46' «9,  X  =  —  3°  io'«9,  the  results  will  exactly  correspond 
with  the  special  calculation. 

Note. — The  expression  of  cos  w  for  the  annular  phase,  as  the  appearance  of  this 
phase  is  comprised  within  narrow  limits  on  the  surface  of  the  earth,  will  afford  a 
very  convenient  and  simple  determination  of  the  places  which  range  in  those  lim- 
its as  well  as  those  which  range  in  the  central  line ;  and  we  may  expect  very  ac- 
curate results  throughout  the  portion  of  country  originally  taken  into  considera 
tion.  Thus  for  the  southern  limit  we  must  obviously  have  cos  w  =  -J-  i,  for  the 
central  line  cosw  =  o,  and  for  the  northern  limit  cos  u  = — i;  and  hence  the 
following  conditions: 

(  +  i  }          I  southern  limit. 

p  —  L'  sin  /  -f-  /  cos  /  cos  (A  -f  «//')  =  <       O>  for  <  central  eclipse. 

(  —  i  )          (  northern  limit. 
By  making  the  assumptions 

ri  cos  2F  =  y'  cos  (A  -f  i//') )  f 

ris\nN'  =  L'  J 

*hey  will  give 

f — p  +  i  ^          (  southern  limit  ^ 

ri  cos  (N'  •+-  /)  =  <  — p          >•  for  <  central  eclipse  >      ....(*) 

(  —  p  —  i  )          (  northern  limit  ) 

If  we  therefore  take  any  meridian  whose  east  longitude  is  A,  these  two  equa- 
tions (r),  («)  will  serve  to  determine  the  extreme  latitudes  /,  on  this  meridian,  be- 
tween which  the  eclipse  will  be  annular  as  well  as  that  where  it  will  be  central 
For  the  preceding  eclipse,  these  equations  will  be 

ri  cos  N1  =  [i  .46950]  cos  (X  4-  i°  4a'  4), 
w'sin  ^'  =  [r.75i59]; 

f  _  [1.45737])          r  southern  limit. 

ri  cos  (2V  +  1)  =    •]  —  [i  -47226]  >  for  <  central  eclipse. 

( —  [z  .48665]  )          (  northern  limit. 


APPENDIX    XI. 

If  we  take,  for  example,  the  meridian  of  Edinburgh,  and  use  A=  —  3°  io'»9, 

there  will  result, 

o      '        » 

Extreme  southern  point  of  annular  appearance,  N.  54  19-7 
Point  of  central  appearance,  N.  55  20 -4 
Extreme  northern  point  of  annular  appearance,  N.  56  21*7 
which  are  geocentric  latitudes. 

III.  CALCULATION  OF  THE  TRANSIT  OF  MERCURY, 

November  7,  1835. 

The  conjunction  in  right  ascension  takes  place  at  out  7h  38"1;  take  therefore 
T=  7h  4om,  and  we  readily  find  from  the  ephemeris  the  following  data: 

i  —  16°  i5'  58"«2 

O          I          II  I        II 

J)  —  l6   22      4«2  a    +      O    10.95 

A  —         2  32-6  a,  +    5  32.7 

9  4-8  v       16  10.4 

P  12.66  ,  8.66 

With  these  quantities,  the  calculation,  for  external  contact  of  limbs,  is  as  follows: 

9    16  10-4 

<  4-8  

A  16  i5«2  4'Qo  .  .  .  o«6owO 

A  2-98909 
6  +  7-61297 

a   -f    I* 03941  ai   +   2-52205 

cos  £  +  9-98226 +  9.98226 

•  COS  i    +    I.02I67  «jC08<J    +    2«5o43l 

— 1°6  i558-2)  A  —   232-6    .    .    —2-18355 

•O  )  a,  COS  6   +    2-5o43l 

_4«_2      a  COS  3    +   1-02167 

"6-~o  .  .   +  2-56348 


tan  ^  +  8^5879"  {  cos  +  9- 9^35 

A   2 • 08009 

•n.f.,.,.,998,  ^   ^gJ2 

+  2'56366  +  6-50074 
A   2-08909 

,+1333  *  + 3.99643 

y  +  9-57457  Jfc  +  3-99643 

cos  ($  +  0   -f  9 •  9^109    sin  ($  +  i)—  9-60749  kb  +  1-60940 

cos  w  +  9^53506                     -3.i7849  8in >W  +  9'97^8 

h.  m.     ».  *"  +  i-6366i 

g  —  o  25     8-3  cos^  +  9.98226 

r+Ul__  *"co8^  +"i-  61888 


sin  w  +  9-97278        T — q  -f  8     5     8-3 
k  +  3-99G43 

I  »n  w  +  3*9(1921    .     .     .        2  35  1 5. 6 


Mean  time  of  j  "J^    f>  ^  J'7  |  for  ^  ^^  rf  the  eartju 


420 


SPhERICAL    ASTRONOMY. 


ffin 


CONSTANTS  FOR  REDUCTION  OF  INGRESS. 
h.  m.    s. 
5  29  52.7 

Equa.  4-       1 6  10-0 

{time  +  5  46     2.7 
arc     +    86°3o'.7 


—  i+    25   32  .3 
w        69   55  -4 


__!_w_  44   23 


—  H  —  io5   58  -3 


cos  +  9«854io 
sin  S  —  9  .44?33 


y  —    19  27  '6 

*''  cos  i  +  i. 61888 
L"  +  1.47298 

CONSTANTS  FOR  REDUCTION  OF  EGRESS. 
b.    m.    s. 
10  40  23.9 
Equa,  +         16     9-2 


sin  —  9.84477 

.      __9.3OI43 

j  tan  +  0.54334 
\  sin   —  9.98290 

-f  9-86187 
k"  +  1-63662 

y"  +  1.49849 


{time  + 

10  56 

33 

.  I 

arc  -f 

164° 

8' 

.3 

-  1  +  W  + 

95 

27 

•7   - 

.  cos  —  8 
sin  S  —  9 

•77854  .   . 
•44733 

.  sin  + 

9' 

9980* 

+  8 

•42587  .   . 

.   .   + 

8. 

42587 

a"  H  + 

88 

„ 

•  o 

j  tan  + 

i- 

572r5 

T      •"  ' 

(   i 

,. 

?r 

• 

9 

99904 

H- 

107 

23 

•7 

k"  cos  6  +  i 

•61888 

k"  + 

9' 

99818 
63662 

L"  — 0-59742 


i-6348o 


The  former  part  of  the  calculation  repeated  for  the  times  5h  3om  and  ioh  4om  we 
shall  find  more  accurate  times  of  ingress  and  egress,  for  the  centre  of  the  earth, 
to  be  5h  29"™  56»  and  ioh  4om  3i",  which,  however,  still  cannot  be  depended  on 
within  a  few  seconds  More  reliance  can  be  placed  in  the  amount  of  reduction  for 
parallax.  The  times  reduced  for  any  place  whose  north  latitude  is  I,  and  east 
longitude  X,  viz. : 

Ingress,  Nov.  7(1  5*  29""  568+  [i  -473o]  p  sin  /  -  [t  -4985]  p  cos  I  cos  (A  —  19°  28') 
Egress,  "  "  10  4o  3t  +[0-5974]  p  sin  I  +  [i  -6348]  p  cos  /  cos  (X  -  107°  24') 
will  indicate,  with  considerable  accuracy,  the  difference  between  the  times  at  any 
two  places. 

The  positions  of  the  contacts  on  the  sun's  limb,  for  an  inverted  image,  will  be 

(  ingress 44°  23'  i  (  west. 

Contact  at  ]  ^   ^      ....     ,5    28  \  ^om  the  north  towards  the  ] 


APPENDIX   XI. 


422 


IV. — OCCULTATION  OF  A  STAR. 

On  January  7, 1836,  the  star  «  Leonis,  whose  right  ascension  is  ioh  2Jm  a6*'4  ami 
declination  N.  i4°  58'  89",  will  be  occulted  by  the  moon. 

LIMITS  OP  LATITUDE. 

At  the  time  of  true  c5  in  right  ascension,  viz.,  i2h  I2m  17",  we  have  the  follow- 
ing data : 


D  +  i5  33     2 
t  +  i4  58  39 
D  —  &+    o  34  23 
with  which  we  proceed  thus: 

Dt—  ii  47       .     —2-84942 
a,  +  3o  4i       .      +  3-265o5 


Di  —  ii  47 

a,  +  30  4l 

P  +  56    4 


to 


<5  +  14059' 


o      ' 
+  1  47  24 


—  9- 58437 
cos  +  9.98498 

tan  —9.59939     w2  +  107  18 
+    21  4i 


!tan  — 909939 

cos  +  9-96813 


const, 
nat  cos 
nat.  cos 


•2725 


diff.  dec.  +  34'  23" 

P  +  56'   4" 
I  +  -5699 


+  3-3i45o 

n  +  3.28263 
+  3.52686 


86  37 


63  5i 


•  8424 

•2974 

log.  cos  +  9-9681  (i) 
log.  cos  +  8-8833  (2) 
log.  cos  <5  +  9-9850  (3) 

log.  cos     +  9.9531  (i)  +  (3) 


9.75577 


+    83  33 
,—      4  1 4  —  log.  sin  li  +  8-8683  (2)  +  (3) 


The  star  may  therefore  be  occulted  between  the  parallels  of  latitude  N.  83°  33' 
and  S.  4°  '4.  The  parallel  of  Greenwich  is  within  these  limits;  and  if  the  hour 
angle  of  the  star  be  computed  roughly  for  the  meridian  of  Greenwich,  the  star  will 
be  found  to  be  considerably  elevated  above  the  horizon.  A  special  calculation  for 
the  observatory  of  Greenwich  will  consequently  serve  as  an  <-xumple  of  the  «i.-- 
curastances  for  a  particular  place. 

CALCULATION  FOR  GREENWICH  OBSERVATORY. 
Constants  f  >,  0  •>,  0(3). 

P   .     .     .     .         9'999l3 
cos  I   .      .      .      +  9.79610 

f")         .        .        .        +   9.79523        .... 

cot  I  .      .      .      +  9.90381     const.    . 

0'2>      .      .      .      +  9-89142     f'«)        .      . 
These  will  be  constant  for  all  occultations  at  Greenwich. 

h.    in.    s. 

Sidereal  time  at  mean  noon  .  19    4  22.4 

Star's  right  ascension       .      .  10  23  26*4 

h  at  mean  noon     . 
Mean  time  of  true  (5 
Acceleration   .... 

h  at  true  6 


+  9.79523 

9.41916 

+  9-21439 


i5 

19    4." 

• 

.     — 

i5 

'9 

4- 

0 

12 

12 

T 

.      .      . 

.      + 

1  1 

6 

2 

acceleration 

.      + 

i 

4y 

4 

—   35 


,.j 


time 


arc 


—  4  ii   14. ft 

—  62°  48'. 7 
With  this  and  a,  =  3o'«7  we  find,  by  the  table  at  p.  405,  T=  nh  6m. 

h  at  mean  noon  is  put  down  negatively,  in  order  to  have  more  readily  the  othor 
values  of  h  lees  than  I2h  or  180°. 


SPHERICAL   ASTRONOMY. 


P  56'  4" 

.     +3.52686   .      .      . 

+  3.52686 

.     .     .    +3.52686 

1P>.     . 

.     +  9-89142     cos  h 

+  9-65983 

sin  h       —  9.94915 

+  3-41828 

+  3.18669 

—  3  47°'01 

cos  a    . 

.     +9.98499     sin  3    . 

+  9.41236 
+  2.59905 

sin  6   .    +  9«4i236 
—  2^88237 

•  »'.-•• 

1       7,     0(1) 

+  9.79523 

0;3>      -    4-  9-21439 

+      48... 

+  2-39428 

—  2-10276 

+    38     3 

—      2'    7" 

D  —  i  . 

.     +   47-22 

Di      .    —    1  1  4a 

t     .     > 

.     +     9  19 

Xl       .    —     9  35 

im 

33'  547' 

|  .,  .    +   3o'  44" 

—  3-3o835 

{           =3-26576 

cos  J      . 

.     +9-98499 

cos  a  .    +9.98499 

1 

j       3-29334 

j  +  3.25075 

1  —32'  45" 

1+    29'  4i" 

JP  sin  h 

.     —3.47601     P     . 

.     3.52686 

PCOS/4    +3.18669 

.     +9.79523     const. 

.     9.43537 

^,(3)       „       +  9.21439 

t  —3.27124      A. 

2*96223 

f   +  2.40108 

1—   3i'    7" 

(                              —I.99I23 

x     .      .      .     +2-  7474i 
tan  S    .      .     —9-24382     8  ^      f 
cos  (S.    .      .     -f  9'99343     , 

o     ' 
—   9  56«6    cot  i  . 
—  20  36«6    cos  t  . 

—  2-75967 

—  0.42474                   .  , 
+  9.97128                      ; 

\ir                           4.  2«753o8 

.     .     .     .    W      . 

+2-  75398                              J 

«)  +  3o  33.2 

3.55630.    .:.. 

n               .        .       +  2.68906 

+  6-28i56 

A'    ...              2.96223 

H      . 

+  3.09715 

COSu,       .        -       +9.72683    .    w          . 

+  57  47  -o  .  cos  w 

+  9.72683 

«      .      .      .     +3.37032       a 
wn  a      .      .     —9-66045       6 

—  27  i3.&     c  .      . 
+  88  20.  2      sin  6  . 

+  3-37032 
+  9.99982 

—  3-o3o77 

+  3-37014 

h        m 

, 

+  oh39m.i 

7*    ...         ii     6 

T  

n     6 

Immersion            10  48  .1      . 
Acceleration                i  «8 
8  T.  mean  noon  19    4  -4 

Emersion        . 
Acceleration 
S.  T.  mean  noon 

ii  45    i  mean  timus. 

2  -0 

19    4  -4 

Immersion              5  54  -3 
Star's  R.  A.         10  23  -4 

.     Emersion       .      • 
Star's  R.  A.    .      . 

6  5  1  «5  .  sid.  tinue^ 
10  23-4                     ' 

j!m.A     .     —4     29.  i  =—67 
"1  Parallactic/  —39°.  7 

0         j  Em,  /*  =    .      . 
(  Parallactic  Z 

—  3  3i  .9  =  —  53" 
-T~3~6°^ 

(  1^    .        .        .        +20-6 

u>    .      .      .     +  57     8 

From|nol'fl      ~"37     J|tothe 
(  vertex    4-2     D  ) 

(-<)•      -      •      • 

(  north 
east.        From  -j 

+      20     -6 

+    57  -8 

+   78  -4)to{he^ 
c    +  ii5  «3  )                f 

APPENDIX    XI. 


423 


These  angles  are  for  the  inverted  image  ;  and,  being  estimated  towards  the  east, 
the  negative  values  must  be  considered  as  towards  the  west.  The  declination  of 
the  star  gives  for  the  latitude  of  Greenwich  a  semi-diurnal  arc  of  7*  a3m ;  as  this 
exceeds  the  value  of  A  both  at  immersion  and  emersion,  the  immersion  and  emer- 
sion will  both  occur  above  the  horizon. 

V. CALCULATION    OF   THE   ECLIPSE   OF   THE   MOON, 

April  30,  1836. 

The  opposition  or  full  mocn  takes  place  at  I9h  58m.  For  the  computation  assume 
the  time  aoh  o™. 


8 


19" 

b.     in.    8. 

20h 
b.    m.    s. 

21" 
b.  m.    s. 

>'sR.  A. 

.     .     14  32  5i-35     . 

.     14  35  11-19 

.     .     14  37  3i-43 

©'s  B.  A.  -\ 

-  i2h     i4  33  52-38     . 

.     14  34    1.91 

.     .     14  34  n-45 

(  time       —     i     i  «o3 

+    i     9-28 

+    3  19-98 

(  space      —      1  5'   i5" 

+     17'  19" 

+     5o'    o" 

a 

=  +  '7    '9     .    32   4l 

"    .,=  +  3,'  38' 

, 

+  5o     o 

i9h 

20>> 

21* 

O        '        " 

O          '       " 

0         '      * 

)  's  dec. 
a  cor. 

.   .  -i4    5  19)  .  "  . 
of 

—  ?4  19  58)  . 

.    —  14  34  32  ) 
5J 

0's  dec. 

.   .    +i5    6  35     .     . 

+  i5     7  20     . 

.     +i5    8    6 

x  . 

.   .    +~i     T~i6 

+        47~27 

+       33^ 

+  6''  I6"_  I3'  55 

H 

i 

P  =  +  47     21                   3     *i 

x\  ^  —  i3'  5^ 

+  33    29 

0  +  3-01662 

ai  +  3.29181 

P  =  60'  19" 

cos  D  +  9-  98627 

.     .     +9-98627 

y  +  3-  00289        t/i  +  3.27808 
z  +  3-45347        xi  —  2-92117 

P  3-  55859 
9.99929 

'       -f'  9° 

{tan  $+  9.54942 

3.55788 

P'  .  6o'i3"> 

»    .          9    ) 

cot  i  —  0.35691 

CO8I  +  9.96163 

—  23 

43  .9            17+3-47916 

.     .      +3.479'6 

a     .    l5  53 

+  0+  4 

i3  -2   .    cos     +9.99882 

3.5563o 

44  29 

n  +  3.  47798 

+  6.99709 

*V  .      44 

External  .    .  A'     3-568o8 
i        +35    38  .7        cos  «  -|-  9.90000 

#+3.71901 
.    .    +9.90990 

45  1  3  SHADOW 

8           1  6    26 

\        -~ 
»       J-39 

25  '5              e  -1-3.80911 
5i  »9         sin  a  —  9-71716 

c  +  3-8o9t  i 
sin6+9-8o685 

,  j  61   39  external 
(  28  47  internal 

-3-52627 

+  3.6i5o6 

«,  -      ^-".o 

<2+    ih8'".8 

Assumed  time    2oh     o 

...     20   o 

Beginning      19      4  «o 

Ending  2  1    8  -8 

Greenwb  mean  times. 

4:24 


SPHERICAL    ASTRONOMY. 


For  the  times  at  any  other  place,  it  will  only  be  necessary  to  take  into  account 
the  difference  of  longitude. 

The  positions  of  the  points  of  contact  on  the  limb  of  the  moon  may  be  deter 
mined  in  the  same  manner  as  those  of  an  occultation,  and  will  here  be  unnecessary. 

As  A'  for  internal  contact  with  shadow  is  less  than  n,  no  internal  contact  cau 
take  place,  and  therefore  the  eclipse  is  only  partial. 

The  contacts  with  the  penumbra  are  to  be  determined  in  a  similar  manner  from 
the  same  values  of  »,  //,  and  will  also  be  unnecessary  here. 

A'  for  external  contact  with  shadow     61'  89" 
n  .  5o     6 


Eclipsed  ....     1 1    33 

which  divided  by  2  s  =  3a'  5z",  gives  o-35i  for  the  magnitude  of  the  eclipse,  the 
moon's  diameter  being  unity. 


APPENDIX    XII. 


EQUATION  OF  EQUAL  ALTITUDES. 

Let  P  be  the  pole,  Z  the  zenith,  S' 
the  place  of  the  sun  in  the  afternoon,  S 
the  place  he  would  have  occupied  had  his 
declination  or  polar  distance  P  S  remain- 
ed unchanged.  Make 

I  =  latitude  of  place    =  90°  —  P  Z ; 
x  =  declination  of  sun  =  90°  —  P  S ; 
a  =  altitude  of  sun       =  90°  —  Z  S ; 
P  —  hour  angle  Z  P  S ; 

then  in  the  triangle  Z  P  S, 

sin  a  =  sin  /  sin  x  -f-  cos  /  cos  x  .  cos  P  ...     (a) 

Differentiating,  supposing  x  and  P  alone  to  vary,  we  have 
d  x .  cos  x .  sin  I  =  cos  I .  cos  x .  sin  P  .  d  P  +  cos  / .  cos  P  .  sin  x  d  .r, 

or 

d  x .  (cos  a? .  sir  I  —  cos  / .  cos  P  .  sin  x)  =  d  P  .  sin  P  cos  /  cos  x  • 

whence 

tan  I        tan  #  \ 
sin  P  ~~  tan  /7 


APPENDIX    XIII. 


425 


Denote  by  <5  the  change  in  declination  from  the  next  preceding  to  the 
next  following  noon  or  change  in  48  hours,  and  by  t  the  interval  in  hours 
between  the  epochs  of  equal  altitudes  in  the  morning  and  afternoon.  Then 

48  :  S  :  :  t  :  dx, 
whence 


also 


which  substituted  in  Eq.  (6)  give 

6  .  tan  I  .  t 


_ 

~ 


S  .  tan  x  .  t 


48.siu  7£  t       48  .  tan  7J  t1 

converting  both  members  into  time  and  taking  one  half,  we  have,  after 
writing  d  for  #,  and  making  tt  =  \  X  j1 j  •  ^  P  =  j*o  ^  ^> 

5  .  tan  /  .  <  S  .  tan  rf  .  t 


'       1440  .  sin  71  *       1440  .  tan  7£  * 
as  in  the  text,  page  187.  • 


w 


APPENDIX   XIII. 

CORRECTION  FOR  DIFFERENCES  OF  REFRACTION. 

Let  P  be  the  pole,  Z  the  zenith,  and  S  Fi«-  ia 

the  place  of  the  sun  had  the  air  undergone  no 
change,  and  S'  the  place  as  determined  by 
a  change  of  atmospheric  refraction.  Then, 
employing  the  same  notation  as  in  the  pre- 
ceding appendix  and  resuming  its  equation 
(a),  regarding  the  altitude  a  as  referring  to 
the  place  S  and  P  to  the  hour  angle 
Z  P  Sj  we  have,  writing  d  for  a?, 

sin  a  =  sin  / .  sin  d  +  cos  J .  cos  d .  cos  P, 
and  denoting  the  altitude  of  S'  by  a'  and  the  hour  angle  Z  P  S'  by  P1 

sin  a'  =  sin  I .  sin  c?  +  cos  I .  cos  ef  .  cos  P'  \ 
and  by  subtraction, 

sin  a'  —  sin  a  =  cos  / .  cos  d  (cos  P/  —  cos  P) ; 


426  SPHERICAL  ASTRONOMY. 

but  fiin  a'  —  sin  a  =2  sin  i  (a'  —  a)  .  cos  \  (a'  +  a), 

cos  P'  -  cos  P  =  2  sin-J  (P  -  P') .  sin  4  (P'  +  P) ; 

whence  by  substitution, 

sin  £  (a'—  a) .  cos  J  (a'-f  a)  =  cos  I .  cos  d .  sin  1  (P— P') .  sin  \  (P'-f  P), 

and  because  a  and  a'  as  also  P'  and  P  differ  by  very  small  quantities,  the 
above  becomes,  by  transposing  and  dividing, 

p  _  p'  -      K  -  <*)  •  cos  a 

cos  /  .  cos  d  .  sin  P* 

But  denoting  the  refraction  in  the  afternoon  by  r'  and  that  in  the  morning 

by  r,  we  have 

a!  —  a  =  r'  —  r ; 

substituting  and  converting  both  members  into  time,  and  writing  ttt  for  the 
first  member,  we  have 

'  0  i         (/  -  r)  .  cos  a 

"-Tir'cos/.cosd.8in.P' 
*s  in  the  text  at  page  188. 


TABLES. 


TABLE  I. 

Afr.  Ivory's  Mean  Refractions  ;  with  ike  Logarithms  and  their.  Differ- 
ences annexed. 


i    Zenith 
Dist 

Mean 
Refraction. 

Log. 

Diff. 

Zenith 
Dist 

Mean 
Refraction. 

Log. 

Diff. 

o 

>        „ 

o 

i       n 

I 

0       1-02 

o.oo85 

3oi2 

25 

o  27.24 

1-4352 

201 

2 

2.04 

0.3097 

i763 

26 

28-49 

i-4547 

i95 

-  Q.. 

3 

3.06 

0.4860 

1252 

27 

29.76 

i-4736 

109 

o  c 

4 

4-08 

0-6112 

974 

28 

3i-o5 

1.4921 

i85 

5 

5-  ii 

0.7086 

796 

29 

32.38 

I  -5lO2 

1 

6 

6-i4 

0.7882 

675 

3o 

33.72 

I  .5279 

177 

7 

7.17 

o.8557 

587 

3i 

35.09 

1.5453 

17 

8 

8.21 

0.9144 

5i9 

32 

36-49 

1.5622 

170 

-CO 

9 

9.25 

o.9663 

466 

33 

37-93 

1.5790 

IOO 

C.  I 

10 

io«3o 

i  .0129 

424 

34 

39.39 

i.5954 

164 
_  /?_ 

ii 

11-35 

i-o553 

388 

35 

40.89 

i«6n6 

IO2 

12 

i3 

12.42 

1-0941 
i  «i3oo 

359 
334 

36 
37 

42.42 
44-oo 

i  .62-76 
1.6435 

1  60 

i59 

.  c/; 

i4 

14-56 

i-i634 

3i3 

38 

45.6i 

i  .6591 

1  56 

e  c 

i5 

i5-66 

1-1947 

294 

39 

47-27 

i  .6746 

1  55 

16 

16-75 

1-2241 

278 

4o 

48.99 

1.6901 

1  55 

.r  / 

17 

17-86 

i  -2519 

265 

4i 

5o.75 

i  .7055 

ID4 

18 

18:98 

i-  2784 

252 

42 

52.57 

I  «72OT 

i5a 

'9 

20-  1  1 

i.3o36 

241 

43 

54-43 

1.7358 

i5i 

20 

21-26 

1.3277 

230 

44 

56-35 

i.75io 

r  - 

21 

22.42 

1.3507 

222 

45 

58-36 

1.76611 

IDI 

22 

23-6o 

1.3729 

2l5 

46 

i     o«43 

1-78123 

l5l2 

23 

24*80 

i.3944 

207 

47 

2.57 

1.79637 

i5i4 

M 

o  26*01 

i.4i5i 

48 

i     4-8o 

r.8u55 

< 

428 


SPHERICAL    ASTRONOMY 
TABLE  L— (Continued.) 


Zenith 
Dist 

Mean 

liefruelioii. 

Log. 

Diflf. 

Zenith 

Dist 

Mean 

Refraction. 

Log. 

Diff. 

0         ' 

/         a 

o       ' 

/            ; 

49    o 

I      7-II 

1-82678 

i523 

72    3o 

3     3-23 

2-26299 

429 

5o    o 

9-52 

1-84208 

i53o 

4o 

5-o6 

2-26732 

433 

5i     o 

I2-O2 

1-85747 

i539 

5o 

6-93 

2-27l68 

436 

52       O 

14-64 

1.87298 

i55i 

73  oo 

8-83 

2-276o8 

44o 

53    o 

17-38 

1.88863 

1  565 

IO 

10-77 

2-28o5l 

443 

54    o 

20-24 

i  .  90440 

1  577 

20 

I2-74 

2-28498 

447 

55    o 

23-25 

1-92036 

1  596 

3o 

i4-?5 

2-28948 

45o 

56    o 

26-41 

i-93653 

1617 

4o 

16-80 

2-29402 

454 

57    o 

29-73 

1-95291 

1  638 

5o 

18-88 

2-29860 

458 

58    o 

33-23 

1-96955 

1  664 

74  oo 

21  -OI 

2-3o322 

462 

59    o 

36-93 

1.98646 

1691 

IO 

23-i8 

2  -30789 

467 

60    o 

4o-85 

2-oo368 

1722 

20 

25-39 

2-31259 

470 

61     o 

45-01 

2-02124 

i756 

3o 

27-66 

2-3i734 

475 

62     o 

49-44 

2-03918 

i794 

4o 

29.95 

2-32213 

479 

63    o 

54-17 

2-05754 

1  836 

5o 

32-3o 

2-32696 

483 

64    o 

59-22 

2-o7635 

1881 

75  oo 

34  -70 

2-33i84 

488 

65    o 

2    4-65 

2-09567 

1932 

10 

37-i6 

2-33677 

493 

66    o 

io-48 

2-n555 

1988 

20 

39-65 

2-34i74 

497 

67    o 

i6-78 

2-i36o3 

2048 

3o 

42-21 

2-346-76 

502 

68    o 

23-6i 

2-15719 

2116 

4o 

44-82 

2-35i83 

5o7 

69    o 

3i-o4 

2-17910 

2191 

5o 

47-48 

2-35695 

5l2 

70  oo 

39-i6 

2-20185 

2275 

76  oo 

5o-2I 

2-36212 

5i7 

IO 

4o-59 

2-20573 

388 

10 

53-00 

2-36735 

523 

20 

42-04 

2  •  20063 

39o 

20 

55-85 

2-37263 

528 

3o 

43-52 

2-  2  i  356 

393 

3o 

58-76 

2-37796 

533 

4o 

45-02 

2-21752 

396 

4o 

4    'i-74 

2-38334 

538 

5o 

46-53 

2'22l5o 

398 

5o 

4-79 

2.388-79 

545 

71  oo 

48-  08 

2.22552 

402 

77    00 

•7-91 

2.39430 

55i 

10 

49-65 

2.22956 

4o4 

IO 

ii  -ii 

3.39987 

557 

20 

5i.25 

2.23363 

4o7 

2.O 

14-39 

2-4o5f»o 

563 

3o 

52-87 

2-23773 

4io 

3o 

i7-74 

2-41119 

569 

4o 

54-53 

2.  24l86 

4i3 

4o 

21  -19 

2.41695 

576 

5o 

56.21 

2.24603 

4i7 

5o 

24--2 

2.422-8 

583 

72  oo 

57.92 

2-25O22 

419 

78  oo 

28-0-3 

2.42867 

589 

IO 

59-66 

2-25445 

423 

10 

32.o4 

2-43463 

596 

20 

3     1-43 

2-25870 

'  425 

20 

4  35-84 

2-  44066 

6o3 
1 

TABLES. 
TABLE  I.— (Continued.) 


Zenith 
Dist 

Mean 
Refraction 

Log. 

Difl. 

Zonith 
Dist 

Mean 
Retraction. 

Log. 

Diff. 

0         ' 

78  3o 

4  39-75 

2-44677 

611 

o       ' 

84  20 

8  55-25 

2-72856 

1069 

4o 

43.76 

2-45295 

618 

3o 

9    8-88 

2-  73o43 

1092 

5o 

47-88 

2.45921 

626 

4° 

23-16 

2.75o63 

iu5 

79  oo 

52-12 

2-46556 

635 

5o 

38-12 

2-76202 

r,39 

10 

56.47 

2-47198 

642 

85  oo  |        53-84 

2-773^7 

n65 

20 

5    0-94 

2-47848 

65o 

JO 

ic  io-35 

2-78558 

1  191 

3o 

5-54 

2-48507 

659 

20 

27-73 

2-79777 

1219 

4o 

10-28 

2-49176 

669 

3o 

46-  o3 

2-81025 

1248 

5o 

i5-i6 

2-49853 

677 

40 

ii     5-3o 

2-82302 

1277 

80  oo 

20-19 

2-5o54i 

688 

5o 

25-66 

2-836i  i 

1  309 

10 

25-36 

2-5l237 

696 

86  oo 

47-i5 

2-84951 

1  34o 

20 

30-70 

2-5i944 

707 

10 

12     9-88 

2-86325 

i374 

3o 

36-20 

2.52660 

716 

90 

33-97 

2.87735 

i4io 

4o 

4i-88 

2-53387 

727 

-     3o 

59-  5i 

2-89182 

1  447 

5o 

47'74 

2-54i25 

738 

4o 

i3  26-61 

2-90666 

1  484 

81  oo 

53-79 

2-54874 

749 

5o 

i  3  55-4o 

2  •  92  1  89 

1  523 

10 

6    0-04 

2-55635 

761 

87  oo 

i4  26-04 

2.93754 

1  565 

20 

6-5o 

2-56407 

772 

10 

i4  58-71 

2-95362 

1608 

3o 

i3-i8 

2-57192 

785 

20 

i5  33-6o 

2-97016 

1  654 

4o 

20-09 

2.57989 

797 

3o 

16  10-89 

2-98717 

1701  \ 

5o 

27-26 

2-588oo 

811 

4o 

16  5o-8 

3-oo466 

1749 

82  oo 

34-68 

2-59624 

824 

5o 

•1-7  33-6 

3-02267 

1801 

IO 

42-37 

2-60462 

838 

88  oo 

18  19-6 

3-04I22 

i855 

20 

5o-33 

2-6i3i3 

85i 

10 

19    9-0 

3-o6o3i 

i9o9 

3o 

58-59 

2-62179 

866 

20 

2O      2-2 

3-07998 

1967 

4o 

7    7-19 

2-63o62 

883 

3o 

2O   59'6 

3-  10024 

2026 

5o 

i6-i3 

2-63961 

899 

4o 

22       1-7 

3-i2ii3 

2089 

83  oo 

25-4o 

2-64875 

914 

5o 

23    8-9 

3-14268 

2i55 

10 

35-o5 

2-658o6 

93i 

89  oo 

24    21-8 

3-16489 

2221 

20 

45-10 

2-  66755 

949 

IO 

a5  4o'9 

3-18779 

220X> 

3o 

55-58 

2-67722 

967 

20 

27     7-1 

236l 

4o 

8    6-5o 

2-68708 

986 

3o 

28  4o-8 

3-a3574 

2434 

5o 

17-90 

2-69714 

1006 

4o 

30    23-2 

3.26o83 

2509 

84  oo 

29-80 

2-70740 

1026 

5o 

32  i5-o 

3-28667 

2584 

10 

8  42-24 

2-71787 

io47 

90  oo 

34  17-5 

3-5i334 

2667 

SPHERICAL    ASTRONOMY. 


TABLE  II. 

ifr.  Ivonfs  Refractions  continued :  showing  the  logarithms  of  the  correc- 
tions, on  account  of  the  state  of  the  Thermometer  and  Barometer. 


Thermometer. 

Barometer. 

Logarithm. 

Logarithm. 

Logarithm. 

0 

o 

in. 

80 

9.97237 

5o 

O-OOOOO 

3i-o 

O-OI424 

79 

9.97326 

49 

0.00094 

3o«9 

0.01248 

78 

9.97416 

48 

0-00190 

8 

O-O1143 

77 

9.97506 

47 

O-OO285 

7 

O-OIOO2 

76 

9.97596 

46 

o«oo38o 

6 

0-00860 

75 

9.97686 

45 

0-00476 

5 

0-00718 

74 

9.97777 

44 

o«  00572 

4 

0-00575 

73 

9.97867 

43 

0-00668 

3 

0-00432 

72 

9-97958 

42 

0-00764 

2 

0-00289 

7i 

9.98049 

4i 

0-00861 

I 

000145 

70 

9.98140 

4o 

0-00957 

3o'0 

0-00000 

69 

9.98231 

39 

o«oio53 

29-9 

9.  99855 

68 

9.98323 

38 

o«oii5i 

8 

9-99709 

67 

9.98414 

37 

0-01248 

7 

9.99563 

66 

9.98506 

36 

o-oc346 

6 

9.99417 

65 

9.98598 

35 

o-oi444 

5 

9.99270 

64 

9.98690 

34 

o«oi54i 

4 

9.99123 

63 

9.98783 

33 

0-01640 

3 

9.98975 

62 

9.98875 

32 

0-01738 

2 

9.98826 

61 

9.98969 

3i 

0-01837 

I 

9-98677 

60 

9.99061 

3o 

0-01935 

29-0 

9.98528 

59 

9-%99'54 

29 

o-o2o33 

28-9 

9.98378 

58 

9.99248 

28 

o«o2i33 

8 

9-98227 

57 

9-9934i 

27 

0«O2232 

7 

9-98076 

56 

9.99434 

26 

-  -0233i 

6 

9-97924 

55 

9.99529 

25 

0-02432 

5 

9.97772 

54 

9.99623 

24 

o-o253i 

4 

9.97620 

53 

9.99717 

23 

o-O263o 

3 

9.97466 

52 

9.99811 

22 

0-02730 

2 

9-973i3 

5i 

9.99906 

21 

0-02832 

I 

9.97158 

5o 

O'OOOOO 

20 

0-02933 

28-0 

9.97004 

TABLES. 


4:31 


TABLE  III. 

Mr.  Ivortfs  Refractions  continued :  showing  the  further  quantities  by 
which  the  refraction  at  low  altitudes  is  to  be  corrected,  on  account  of 
the  state  of  the  Thermometer  and  Barometer. 


Zenith 
Distance. 

T 

B 

Zenith 
Distance. 

T 

B 

o      ' 

„ 

o       ' 

„ 

„ 

75    o 

—  0*009 

86  3o 

—  o«3i7 

+  o-5i 

76    o 

0*012 

86  4o 

0-345 

o-56 

77     o 

o-oi5 

86  5o 

0-376 

0-62 

78     o 

0-018 

87    o 

o«4io 

0-68 

79    ° 

O'O23 

tf 

87  10 

0.448 

0-75 

80    o 

o-o3o 

+  o-o4 

87  20 

0*490 

o-83 

81     o 

o-o4o 

o«o5 

87  3o 

0-538 

0-91 

8r  3o 

o-o46 

0*07 

87  4o 

0.593 

I«OI 

82     o 

o-o53 

0-08 

87  5o 

0-654 

i-i3 

82  3o 

o-o63 

O«IO 

88    o 

0.722 

1-26 

83    o 

0-074 

O'll 

88  10 

0-799 

i-4i 

83  3o 

0.089 

o.i3 

88  20 

,0-887 

i-59 

84    o 

0-107 

0.16 

88  3o 

0-987 

1-79 

84  3o 

o«  i3o 

O«2O 

88  4o 

I-IOI 

2-02 

85    o 

O'i59 

0-25 

88  5o 

I-23l 

2.29 

85  10 

0.171 

O«26 

89    o 

I.38o 

2.61 

85  20 

0-184 

0-28 

89  10 

i-55i 

2.98 

85  3o 

0.198 

o.3i 

89    20 

1-749 

3-4i 

85  4o 

0-2l3 

0-33 

89  3o 

1-977 

3.93 

85  5o 

0-229 

0-36 

89  4o 

2-24l 

4-54 

86    o 

0-248 

o-39 

89  5o 

2-549 

5-26 

86  10 

0.269 

0-43 

90    o 

-2-909 

+  6-12 

86  20 

—  0*292 

+  0.47 

The  column  marked  T  is  to  be  multiplied  by  (t  —  50°);  and  the  column  marked 
B  is  to  be  multiplied  by  (b  —  3oin-oo).  The  results  are  to  be  applied  to  the  ap- 
proximate refraction  obtained  by  the -preceding  tables. 


432 


SPHERICAL    ASTRONOMY. 


TABLE  IV. 

For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


I  nter  va 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  U 

h.  in. 

h.  m. 

h.  m. 

2   O 

7.7297 

7.7146 

3  0 

7-7359 

7-7oi5 

4  o 

7.7447 

7.6823 

2 

.7298 

.7143 

2 

•  7362 

•7OIO 

2 

•745i 

•  68i5 

4 

•  73oo 

•7139 

4 

•  7364 

•7oo5 

4 

•7454 

•  6807 

6 

•  73o2 

•  7i36 

6 

.7367 

•  6999 

6 

•  7458 

•  6800 

8 

•73o4 

•7l32 

8 

•7369 

•6993 

8 

.746i 

•  6792 

10 

•  73o5 

•  •7128 

10 

•7372 

•6988 

10 

•7464 

>6784 

12 

•  73o7 

•7I25 

12 

•7374 

.6982 

12 

•  7468 

•  6776 

i4 

.7309 

•7I2I 

i4 

•7377 

.6976 

i4 

•7472 

•  6768 

16 

.73n 

•7117 

16 

-738o 

•6970 

16 

•7475 

•6759 

18 

•  73i3 

•7Tl3 

18 

•  7383 

.6964 

18 

•74/9 

•  675i 

20 

.73i5 

.7io9 

20 

•  7386 

•  6958 

20 

.•7482 

•  6743 

22 

.73r7 

.7io5 

22 

•  7388 

•  6952 

22 

•  7486 

•  6734 

24 

•73i9 

•710l 

24 

•7391 

•  6946 

24 

••7490 

•  67a6 

26 

•732I 

•709-7 

26 

•7394 

•  6940 

26 

.7494 

•  67i7 

;  28 

•7323 

•7092 

28 

•  •739-7 

.6934 

28 

•7497 

•  6708 

3o 

.7325 

•7088 

3o 

•  74oo 

.692-7 

3o 

•75oi 

•  67oo 

32 

•7327 

•7083 

32 

.74o3 

.6921 

32 

•  75o5 

•  6691 

34 

•7329 

•  7079 

34 

.74o6 

-6914 

34 

•  75o9 

.6682 

36 

.733i 

•7075 

36 

•74o9 

•  6908 

36 

•  75i3 

•  6673 

38 

•  7333 

•7070 

38 

•74l2 

•  6901 

38 

•75i7 

•  6663 

4o 

•  7336 

•7065 

4o 

•74i5 

.6894 

4o 

.752I 

•  6654 

42 

•  7338 

•7061 

42 

•  74i8 

.6888 

42 

•  7525 

•  6645 

44 

.734o 

•7066 

44 

•742I 

•  6881 

44 

.7529 

•  6635 

46 

•  7342 

•  7o5i 

46 

.7424 

•  6874 

46 

•  7533 

•  6626 

48 

•7345 

•7046 

48 

.•7428 

.686-7 

48 

•  7537 

•  6616 

5o 

•7347 

•  7041 

5o 

•743  1 

.6859 

5o 

•754i 

.6606 

52 

•7349 

.7036 

52 

-7434 

.6852 

52 

•7545 

•6597 

54 

•  7352 

•  7o3i 

54 

•7437 

.6845 

54 

•7549 

.6587 

56 

•7354 

•  7026 

56 

r?44i 

.6838 

56 

•  7553 

•6577 

58 

7.7357 

7.7021 

58 

7-7444 

7-683o 

58 

7.7557 

7-6567 

. 

i 

TABLES. 


433 


TABLE  IV.— (Continued.) 
For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


Interval 

Log.  A. 

Log.B. 

Interva 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

h.  m. 

h.  m. 

b.  m. 

5  o 

7.7562 

7.6556 

6  o 

7-7703 

7.6198 

7  o 

7.7873 

7.5717 

2 

•  7566 

.6546 

2 

.7708 

•6184 

2 

•7879 

.5699 

4 

.7570 

.6536 

4 

-77*3 

•  6170 

4 

•  7885 

•  568o 

6 

•7575 

•  6525 

6 

.7719 

•  6i56 

6 

.7891 

•  566  1 

8 

•7579 

•  65i4 

8 

.7724 

•  6142 

8 

.7898 

•  564  1 

10 

•7583 

•  65o4 

10 

.7729 

•  6127 

10 

.7904 

•  6622 

12 

•  7588 

•  6493 

12 

.7735 

•  6n3 

12 

.7910 

.5602 

i4 

.7692 

•  6482 

i4 

.774o 

.6098 

i4 

.7916 

•  5582 

16 

•7597 

•  -6471 

16 

•7745 

•  6o83 

16 

.7923 

.5562 

18 

•  76oi 

•  646o 

18 

•775  1 

•  6068 

18 

.7929 

•  5542 

20 

•  76o6 

•  6448 

20 

•7756 

•  6o53 

20 

•7936 

•  5522 

22 

.76io 

.6437 

22 

.7762 

•6o38 

22 

.7942 

•  55oi 

24 

•76i5 

•  6425 

24 

•7767 

•  6023 

24 

•7949 

.5480 

26 

•  •7620 

.64:4 

26 

•7773 

-6007 

26 

•  7955 

•  5459 

28 

•  7624 

•  6402 

28 

•7779 

•5991 

28 

•  7962 

•5437 

3o 

•  7629 

.639o 

3o 

•7784 

.5975 

3o 

•7969 

•  54i6 

32 

•7634 

•  6378 

32 

.7790 

•5959 

32 

.7975 

•  5394 

34 

•  7638 

.6366 

34 

.7796 

•5943 

34 

.7982 

.5372 

36 

•  7643 

.6354 

36 

.7801 

•5927 

36 

.7989 

.535o 

38 

•7648 

•  6342 

38 

.7807 

•5910 

38 

•  7995 

•  5327 

4o 

•  7653 

.6329 

4o 

.78i3 

•5894 

4o 

.8002 

.53o4 

42 

•  7658 

.63i7 

42 

.7819 

.5877 

42 

•  8009 

.5281 

44 

•  7663 

-63o4 

44 

.7825 

•586o 

44 

.8016 

.5258 

46 

.7668 

•6291 

46 

•  783  1 

•  5843 

46 

•  8023 

.5234 

48 

•7673 

.6278 

48 

•  7836 

•  5825 

48 

•  8o3o 

.5211 

5o 

.7678 

.6265  9 

5o 

.7842 

•58o8 

5o 

•  8o37 

•  5i86 

52 

-7683 

-6252 

52 

•  7848 

•5790 

52 

•  8o44 

•  5i62 

54 

.7688 

.6^39 

54 

.7854 

•5772 

54 

•  8o5i 

•  5i37 

56 

.7693 

•  6225 

56 

•  7860 

•5754 

56 

•  8o58 

•5lI2 

58 

7.7698 

7.6212 

58 

7.7867 

7.5736 

58 

7-8o65 

7.5087 

• 

9.S 


434 


SPHERICAL   ASTRONOMY. 


TABLE  IV.— (Continued.) 
For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.B. 

h.  m 

h.  m. 

h.  m. 

8  o 

7.8072 

7.5062 

9  ° 

7-8302 

7-4i3i 

IO  0 

7.8567 

7.2697 

2 

•8079 

•  5o36 

2 

•  83n 

•  4o93 

2 

•  8576 

•  2635 

4 

•  8086 

•  5oio 

4 

•  83j9 

•4o55 

4 

•  8586 

•2572 

6 

•8094 

.4983 

6 

•8328 

•4oi6 

6 

•  8595 

.2507 

8 

•  8101 

•4957 

8 

•  8336 

.3977 

8 

•86o5 

.2442 

10 

•  8108 

.493o 

10 

•  8344 

.8937 

10 

•  86i4 

•2374 

12 

•  8116 

•  4902 

12 

.8353 

.38o6 

12 

•  8624 

•  23o6 

i4 

•  8i23 

.4874 

i4 

.836  1 

•  3855 

i4 

.8634 

•2236 

16 

•  8i3o 

•  4846 

16 

•  837o 

•  38i3 

16 

.8643 

•  2164 

18 

•  8i38 

.4818 

18 

•  8378 

.3771 

18 

•  8653 

.2091 

20 

•8i45 

.4789 

20 

•  8387 

.3728 

20 

.8663 

.20,6 

22 

•  8i53 

.4760 

22 

.8396 

.3684 

22 

•  8673 

•  ip4o 

24 

•  8160 

•473i 

24 

•84o4 

•  3639 

24 

•  8683 

.1861 

26 

.8168 

•4701 

26 

.84x3 

•  3594 

26 

•  8693 

.1761 

28 

.8176 

•4671 

28 

•  8422 

•  3548 

28 

•  87o3 

.1699 

3o 

.8i83 

•464o 

3o 

•  843o 

•  35oi 

3o 

.87i3 

•  i6i5 

32 

•8191 

•4609 

32 

-8439 

.3454 

32 

•  8723 

•  1529 

34 

.8199 

•4578 

34 

.8448 

.34o6 

34 

•  8733 

.i44o 

36 

•  8206 

•4546 

36 

•  8457 

•  3357 

36 

•  8743 

.1349 

38 

•  8214 

•45i4 

38 

•8466 

.3307 

38 

•  8753 

•  1256 

4o 

•  8222 

.4482 

4c 

•  8475 

.3256 

4o 

•  8763 

.1160 

42 

•  823o 

•4449 

42 

.8484 

.32o5 

42 

•  8773 

.1061 

44 

.8238 

•44i5 

44 

.8493 

•  3i52 

44 

.8784 

•0960 

46 

8246 

•438  1 

46 

•  85o2 

.3o99 

46 

•8794 

•  o855 

48 

.8254 

•4347  . 

48 

.85n 

•  3o45 

48 

•  88o4 

•  07,48 

5o 

.8262 

•43i2 

5o 

.8520 

.2989 

5o 

•  88i5 

•o637 

52 

•  8270 

.4277 

52 

.853o 

•  2933 

'   52 

•  8825 

•0522 

54 

•  8278 

.4241 

54 

.8539 

.2876 

54 

.8836 

•  o4o4 

56 

•  8286 

.4205 

56 

•  8548 

.2817 

.   56 

.8846 

•  0282 

58 

7.8294 

7.4168 

58 

7.8558 

7.2758 

58 

7.8857 

7-oi56 

TABLES. 


435 


TABLE  IV.— (Continued.) 
For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


1 
Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

Intcrva' 

Log.  A. 

Log.  B. 

i 

b.  in 

h.  in. 

h.  m. 

II  O 

7-8868 

7.0025 

12  0 

7.9208 

£  =  0 

i3  o 

7.9593 

—  7-0750 

2 

•  8878 

6-9889 

2 

.9220 

-5-5549 

2 

-9607 

•0905 

4 

.8889 

•9748 

4 

.9232 

5-8641 

4 

-9620 

•I0r;6 

6 

•  8900 

.9602 

6 

.9M5 

6-o4r4 

6 

-9634 

•I2O3 

8 

•  8911 

•  9449 

8 

.9257 

.1675 

8 

•  9648 

•  i345 

10 

.8922 

.9290 

10 

.9269 

•  2657 

10 

.9662 

•  i484  \ 

12 

.8932 

•  9126 

12 

.9281 

•  346i 

12 

.9676 

•  1619 

i4 

•  8943 

.8953 

i4 

.9294 

•4i42 

14 

.9690 

•i?5i  | 

16 

•  8954 

.8770 

16 

-93o6 

•4?34 

16 

.9704 

•  1880 

18 

.8965 

•  858o 

18 

.9319 

.5258 

18 

.9718 

•  2006 

20 

.8977 

•  8379 

20 

.933  1 

.5728 

20 

-9732 

.2129 

22 

.8988 

.8168 

22 

.9344 

•6i54 

22 

•9746 

.2249 

24 

«8999 

.7945 

24 

.9357 

.6545 

24 

.9761 

.2367 

26 

.9010 

.7709 

26 

.9369 

•  6oo5 

26 

.9775 

•  2482 

28 

•  9021 

•  7457 

28 

.9382 

.7239 

28 

.9789 

2595 

3o 

.9o33 

.7189 

3o 

•  9395 

•755i 

3o 

.9804 

•  2706 

32 

•9°44 

•  6901 

32 

.9408 

.7843 

32 

.9818 

.2815 

34 

.9o55 

•  6591 

34 

.9421 

•  8119 

34 

•  9833 

.2922 

36 

.9067 

•  6255 

36 

.9433 

,  .838o 

36 

•  9848 

-3o26 

38 

.9078 

.5889 

38 

•  9446 

.8627 

38 

.9862 

.3i29| 

4o 

•  9090 

•  5487 

4o 

.9460 

•  8863 

4o 

-9877 

.323i 

42 

•  9102 

•  5o4i 

47 

•9473 

.9087 

42 

-9892 

.333o  1 

44 

•  9ii3 

•454  1 

44 

-9486 

•  9302 

44 

.9907 

.3428 

46 

•  9125 

•3973 

46 

.9499 

•95°7 

46 

.9922 

.3524 

48 

•9l37 

-33i6 

48 

.9512 

.9705 

48 

.9937 

-3619 

5o 

.9148 

.2536 

5o 

.9526 

6.9895 

5o 

-9952 

.37ia 

52 

•  9160 

•  1579 

52 

.9539 

7.0078 

52 

-9967 

.38o4  i 

54 

.9172 

6-o34i 

54 

•  9552 

.0254 

54 

-9982 

.3894 

56 

'91  84 

5.8593 

56 

•  9566 

.0425 

56 

7.9998 

-3o84 

58 

7-9196 

5-5594 

58 

7.958o 

-7-0590 

58 

8-ooi3 

-  7-4071 

SPHERICAL   ASTRONOMY. 


TABLE  IV.— (Continued.) 
For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A.   Log.  B.  I 

h.  m. 

h.  m. 

h.  m. 

14  o 

8-oo-iS 

-  7-4i58 

,5o 

8-o52i 

-  7-635o 

16  0 

8-1082 

—  7.8072 

2 

•  oo44 

•4244 

2 

.o539 

-64i3 

2 

•IIO2 

.8125 

4 

•  0059 

.4328 

4 

•  o556 

•6475 

4 

-1122 

.8177 

6 

•  0075 

•44i2 

6 

•o574 

.6537 

6 

.1143 

.8229 

8 

•  0090 

.4494 

8 

•  0592 

•6599 

8 

•n63 

•  8abi  ; 

10 

•  0106 

•4575 

10 

•  0610 

•6660 

10 

•  ii83 

.8333 

12 

•OI22 

•  4655 

12 

•  0628 

•  6721 

12 

•  1204 

•  8385  ! 

i4 

•  oi38 

••4735 

i4 

•  0646 

•  6781 

i.4 

•  1224 

•  8436 

16 

•  oi54 

•48i3 

16 

.0664 

•6841 

16 

•  1245 

•  8487 

18 

•  0170 

.4890 

18 

.0682 

.6900 

18 

.1266 

.8538 

20 

•0186 

.4967 

20 

•  0700 

.6959 

20 

•1287 

.8589 

22 

•02O2 

.5o43 

22 

.0718 

.7018 

22 

•i3o8 

•  864o 

24 

•02l8 

.5n8 

24 

.0737 

.7077 

24 

•  1329 

•  8690 

26 

•0234 

•  5192 

26 

•  o755 

•  7i35 

26 

•i35o 

•  8740 

28 

•O25O 

•  5265 

28 

.0774 

.7192 

28 

-i37i 

.8790 

3o 

,0267 

•  5338 

3o 

•  0792 

.7249 

3o 

•i393 

•  884o 

32 

•0283 

•  54io 

32 

•  0811 

•  73o6 

32 

•i4i4 

•  8890 

34 

•o3oo 

•  548  1 

34 

.o83o 

•  7363 

34 

•i  436 

.8939 

36 

•o3i6 

.555i 

36 

•0849 

.7419 

36 

•i  458 

•  8989 

38 

•o333 

.5621 

38 

•0868 

•7475 

38 

•i  479 

•  9038 

4o 

•o35o 

.5690 

4o 

•  0887 

•  753i 

4o 

«i5or 

.9087 

42 

•o367 

•5759 

42 

•  0906 

.7586 

42 

•  i5tt3 

•  9i36 

44 

•  o384 

.5827 

44 

•  0925 

.7641 

44 

.i545 

.9i85 

46 

•  o4oo 

.  .68.94 

46 

•  0945 

7696 

46 

•  i  568 

.9234 

48 

•  0417 

.5961 

48 

•  0964 

775  1 

48 

i59o 

.9282 

5o 

•  o435 

•  6027 

5o 

.0983 

•  78o5 

5o 

•1612 

•  933o 

52 

•  o452 

.6092 

52 

«ioo3 

.7859 

52 

•i635 

.9379 

54 

•  0469 

•  6i58 

54 

•1023 

.7912 

54 

•i  658 

.9427 

56 

•  o486 

.6222 

56 

•  1042 

.7966 

56 

•  1680 

.9475 

58 

8-o5o4 

-7.6286 

58 

8-1062 

—  7.8019 

58 

8-  1703 

-  7-9523 

1 

1 

TABLES. 


TABLE  IV.— (Continued.) 
for  the  Equation  of  Equal  Altitudes  of  the  Sun. 


r* 
Interval 

Log.  A. 

Log.B. 

Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

b.  m. 

h.  m. 

b.  m. 

17  0 

8-1726 

-7.9571 

18  0 

8-2474 

-8.0-969 

19  O 

8.3359 

-8-2354 

2 

•1749 

.9618 

2 

.2501 

•  ioi5 

2 

•  3392 

•  2401 

4 

•1773 

.9666 

4 

.2529 

•  1061 

4 

.3^24 

.2448 

6 

•1796     '97l3 

6 

.2556 

•  1107 

6 

•3457 

.2495 

8 

•  1819 

.9761 

8 

.2583 

.ii53 

8 

•,3490 

•2542 

10 

.1843 

.9808 

10 

.2611 

.1199 

10 

•3524 

.2589 

12 

•  1867 

.9855 

12 

-2639 

.1245 

12 

•3557 

•  2637 

i4 

•  1890 

.9902 

i4 

.2667 

.  1291 

i4 

.359l 

-2684 

16 

-1914 

.9949 

16 

.2695 

•  i  336 

16 

.3625 

•  2732 

18 

.i938 

7-9996 

18 

.2723 

.1382 

18 

•  3659 

.2779 

20 

.io63 

8-oo43 

20 

.2752 

•  1428 

20 

-3694 

2827 

22 

.1987 

.0090 

22 

.2781 

•i474 

22 

.3728 

•  2875 

24 

.2011 

•  0137 

24 

.2809 

•l52O 

24 

-3763 

•  2923 

26 

.2036 

.0184 

26 

.2838 

•  i  566 

26 

.3798 

•  2971 

28 

.2061 

.0230 

28 

.2£68 

•  1612 

28 

•  3834 

.3019 

3o 

.2086 

.0277 

3o 

.2897 

-i  658 

3o 

-3869 

•  3o68 

32 

.2111 

•  o323 

32 

.2926 

-1704 

32 

-39o5 

.3xx6 

34 

•  2i36 

.0370 

34 

-  2956 

•1750 

34 

-394i 

-3x65 

36 

•  2161 

•  0416 

36 

.2986 

-1797 

36 

•  3978 

.32x4 

38 

.2186 

•  0462 

38 

.3oi6 

•  1842 

38 

/4ox5 

.3263 

4o 

•  2212 

•o5o8 

4o 

•  3o46 

•  1889 

4o 

-4o52 

.33x2 

42 

•2237 

•o555 

42 

.3077 

•  x935 

42 

.4089 

•  336  1 

44 

.2263 

•  0601 

.  44 

•  3i07 

.1981 

44 

.4126 

.34io 

46 

.2289 

.0647 

46 

•  3x38 

•  2028 

46 

•  4i64 

.346o 

48 

•  23i5 

.0693 

48 

-3i69 

•2074 

48 

.4202 

-35xo 

5c 

•  234i 

•0739, 

5o 

•3200 

•2121 

5o 

•4241 

.356o 

52 

2367 

•o785 

52 

•3232 

•2167 

52 

.4279 

.36io 

54 

•  2394 

•  o83  1 

54 

-3263 

«22l4 

54 

•  43x8 

.366o 

56 

-2420 

•  0877 

56 

.3295 

•226l 

56 

•4357 

•  37ix 

58 

8-2447 

—  8-0923 

58 

8-3327 

—  8-2307 

58 

8.4397 

-8-376i 

SPHERICAL   ASTRONOMY. 


TABLE  IV.— (Continued.) 
For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


Interval 

Log.  A. 

Log.  B. 

[uterval 

Log.  A. 

Log.  B. 

interval 

Log.  A. 

Log.  B. 

h.  m. 

h.  m. 

h.  m. 

2O  O 

8-4437 

-8-3812 

21   0 

8.58io 

-8.5466 

22  O 

8.7711 

-8.756o 

2 

•4477 

.3863 

2 

•  5863 

.5527 

2 

.7789 

•7643 

4 

•  45i8 

•  39i5 

4 

.5917 

.5588 

4 

•  7868 

•7727 

6 

•  4559 

.3966 

6 

•  5971 

•  565o 

6 

•7948 

•  78i3 

8 

.4600 

•  4oi8 

8 

•  6025 

.5712 

8 

•8o3o 

.7899 

10 

.464  1 

.4070 

10 

.6081 

•  5775 

10 

•  8n3 

.7987 

12 

•4683 

«4l22 

12 

•  6i36 

.5838 

12 

•  8198 

•  8076 

M 

.4726 

•4175 

i4 

•  6193 

.5902 

i4 

•  8284 

.8167 

16 

.4768 

.4227 

16 

•  625o 

.5966 

16 

•  8372 

.8259 

18 

•  48n 

.4280 

18 

•  63o8 

.6o3i 

18 

•  846  1 

.8353 

20 

•  4854 

.4334 

20 

.6366 

.6096 

20 

.8553 

.8448 

22 

•4898 

•  4387 

22 

.6426 

.6162 

22 

•8645 

.8545 

24 

•4942 

•444i 

24 

.6486 

.6229 

24 

•874o 

.8644 

26 

.4987 

.4495 

26 

.6546 

.6296 

26 

•8837 

.8745  ' 

28 

•  5o32 

•4549 

28 

.6608 

.6364 

28 

•  8935 

.8847 

3o 

.5077 

.4604 

3o 

•  6670 

.6433 

3o 

•  9o36 

.8952 

32 

•  5i23 

•  4659 

32 

.6733 

.6502 

32 

'9l39 

•  9o58 

34 

.5169 

.47i4 

34 

.6796 

.6572 

34 

.9244 

.9167 

36 

•  52i5 

.4770 

36 

•  6861 

•  6643 

36 

.9351 

..9278 

38 

.5262 

.4826 

38 

.6927 

.67i5 

38 

•  946i 

.9391 

4o 

•  53io 

.4882 

4o 

.6993 

.6788 

4o 

•9574 

.9507 

42 

.5357 

.4939 

42 

.7060 

.6860 

42 

.9689 

.9626 

44 

•  54o6 

.4996 

44 

.7128 

.6934 

44 

.9807 

•9747 

46 

.5455 

•  5c  >3 

46 

.7197 

.7009 

46 

8.9928 

.9871 

48 

•  55o4 

.r.n 

48 

.7268 

.708  5 

48 

9«oo52 

8-9999 

5o 

•  5554 

•  5i69 

5o 

.7339 

.7162 

5o 

•  0180 

9-0129 

52 

•  56o4 

.5228 

52 

«74ii 

.7239 

52 

«o3n 

.0263 

54 

•  5655 

.5287 

54 

-7484 

•  73i8 

54 

.0446 

•  o4oi 

56 

•  5706 

•534t> 

56 

•  7558 

.7398 

56 

•  o585 

•  o543 

58 

8.5758 

-8-54o6 

58 

8.7634 

-  8.7478 

58 

9.0729 

_  9.0689 

TABLES. 


439 


TABLE  IV.— (Continued.) 
For  the  Equation  of  Equal  Altitudes  of  the  Sun. 


Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

Interval 

Log.  A. 

Log.  B. 

h.    m. 

h.    m. 

b.    m. 

23     0 

9-0877 

-  9.0889 

23    2O 

9.2693 

—  9-2677 

23  4o 

9.5761 

-  9-5757 

2 

-1029 

.0995 

22 

.2922 

.2907 

42 

.6224 

•  6221 

4 

•1187 

•  n55 

24 

•  3i62 

•  3i49 

44 

•  6742 

•  6739 

6 

•  i35i 

.1321 

26 

.34:6 

•  34o4 

46 

.7328 

•  7326 

8 

•1620 

•1492 

28 

.3685 

.3674 

48 

«8oo3 

•  8001 

10 

•1696 

•  1670 

3o 

.3971 

.3962 

5o 

.8801 

•  8800 

12 

•  [879 

-i855 

32 

.4276 

•  4268 

52 

9.9776 

9  .9775 

i4 

•  2069 

•  2047 

34 

•  46o4 

•4597 

54 

o-io3i 

o-io3i 

16 

•  2268 

•  2248 

36 

•  4957 

.4952 

56 

0-2798 

0-2798 

18 

9.2476 

-9.2456 

38 

9-5341 

-9-5336 

58 

o-58i4 

-o.58i4 

440 


SPHERICAL    ASTRONOMY. 


TABLE  V. 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
_  2  sin2  *  P 
~~      sin  I77""' 


Sec. 

Om 

lm 

2m 

gm 

4m 

5m 

gm 

7" 

/f 

// 

// 

n 

„ 

„ 

n 

r 

0 

o«o 

2«O 

7.8   ' 

17-7 

3i*4 

49-1 

70-7 

96.2 

I       l 

o«o 

2-0 

8-0 

17-9 

3i-7 

49.4 

71.  1 

96.7 

2 

o-o 

2-  I 

8-r 

18.1 

31.9 

49.7 

71.5 

97.1 

3 

O'O 

2«2 

8-2 

i8«3 

32-2 

5o«i 

71.9 

97-6 

4 

o«o 

2-2 

8-4 

18.5 

32-5 

5o-4 

72-3 

98-0 

5 

0-0 

v   2>3 

8-5 

18-7 

32.7 

5o.7 

72.7 

98-5 

6 

O'O 

2-4 

8-7 

18-9 

33-0 

5i-i 

73-i 

•99.0 

7 

o»o 

2-4 

8-8 

19-1 

33-3 

5i-4 

73.5 

99.4 

8 

o«o 

2-5 

8-9 

I9.3 

33-5 

5i.7 

73-9 

99.9 

9 

o«o 

2-6 

9-1 

19.5 

33-8 

52-1 

74-3 

100-4 

10 

O.I 

2-7 

9-2 

19.7 

34-1 

52  4 

74.7 

100.8 

ii 

O«I 

2-7 

9.4 

19.9 

34-4 

52'7 

75-  1 

101  .3 

12 

O-I 

2-8 

9-5 

20-1 

34-6 

53«i_ 

75-5 

ioi«8 

i3 

O'l 

2.9 

9-6 

2O-3 

34-9 

53-4* 

75-9 

102.3 

i4 

•   o-i 

3-0 

9-8 

20-5 

35-2 

53.8 

76.3 

102.7 

i5 

O-I 

3-1 

9.9 

20-7 

35-5 

54-1 

76.7 

103.2 

16 

O'l 

3-1 

IO-I 

20  -9 

35-7 

54-5 

77.1 

io3'7 

i? 

0-2 

3-2 

IO«2 

21-  2 

36-0 

54-8 

77-5 

io4«  2 

18 

O-2 

3-3 

10  «4 

21-4 

36-3 

55-1 

77-9 

io4*6 

J9 

O«2 

3-4 

10-5 

21-6 

36-6 

55.5 

78.3 

io5«i 

20 

O«2 

3-5 

10-7 

21-8 

36-9 

55-8 

78.8 

io5-6 

21 

O-2 

3-6 

i0'8 

22.  O 

37.2 

56-2 

79.2 

106.1 

22 

0-3 

3.7 

ii  -o 

22-3 

37-4 

56-5 

79.6 

io6'6 

23 

0-3 

3.8 

II  -2 

22-5 

37-7 

56'9 

80.0 

107.0 

24 

o.3 

3-8 

ii.  3 

22.7 

38-o 

57-3 

8o-4 

107.5 

25 

0-3 

3.9 

ii.  5 

22'9 

38-3 

57.6 

80-8 

108-0 

26 

0-4 

4-o 

ii.  6 

23-1 

38-6 

58-o 

8i.3 

io8-5 

27 

0-4 

4-i 

ii-8 

23-4 

38-9 

58-3 

81-7 

109-0 

28 

0.4 

4-2 

ii.  9 

23-6 

39-2 

58.7 

82.1 

109.5 

29 

0-5 

4-3 

12-  I 

23-8 

39-5 

59-0 

82-5 

Ho-o 

TABLES. 


441 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
2  sin2      P 


Sec. 

om 

lm 

2m 

gm 

4m 

5m 

6m 

7, 

tl 

a 

n 

M 

u 

M 

; 

• 

3o 

0-5 

4.4 

12-3 

24'0 

39.8 

59.4 

83-0 

i  io«4 

3r 

0-5 

4-5 

12-4 

24-3 

4o«i 

59-8 

83-4 

110*9 

32 

0.6 

4-6 

12-6 

24-5 

4o-3 

60.  i 

83.8 

111.4 

33 

0-6 

4-7 

12-8 

24.7 

4o-6 

6o-5 

84-2 

in  .9 

34 

0-6 

4-8 

12.9 

25-0 

4o«9 

60-8 

84-7 

II2-4    i 

35 

0-7 

4.9 

i3-i 

25-2 

4l-2 

61.2 

85-i 

112-9 

36 

0.7 

5.o 

i3.3 

25.4 

41-5' 

61.6 

85-5 

u3.4 

37 

0-7 

5.i 

i3-4 

25.7 

4i-8 

61  .9 

86.0 

113.9 

38 

0.8 

5-2 

i3-6 

25.9 

42.1 

62.3 

86-4 

"4-4 

I     39 

0-8 

5-3 

i3-8 

26*2 

42-5 

62-7 

86-8 

n4«9 

4o 

0-9 

5-4 

i4-o 

26.4 

42-8 

63.o 

87-3 

u5-4 

1    * 

0-9 

5-6 

i4-i 

26-6 

43.i 

63-4 

87.7 

115.9 

42 

I  «O 

5.7 

i4-3 

26.9 

43.4 

63-8 

88.1 

116-4 

43 

I»O 

5-8 

i4-5 

27.1 

43-7 

64-2 

88-6 

116.9 

44 

I«I 

5.9 

i4-7 

27.4 

44-o 

64-5 

89-0 

117.4 

45 

I  •! 

6-0 

14.8 

27.6 

44.3 

64-9 

89.5 

117.9 

46 

I«2 

6-i 

i5.o 

27.9 

44.6 

65-3 

89.9 

118-4 

4? 

!•• 

6-2 

l5-2 

28.1 

44.9 

65.7 

90.3 

118-9 

48 

i«3 

6.4 

i5-4 

28.3 

45-2 

66.0 

90-8 

119.5 

49 

i-3 

6-5 

jS-6 

28.6 

45-5 

66-4 

91.2 

I2O«O 

5o 

i-4 

6-6 

i5*8 

.  28.8 

45-9 

66-8 

91.7 

I2O«5 

5i 

i-4 

6-7 

,5.g 

29.1 

46.2 

67-2 

92-1 

121.  0 

52 

i«5 

6-8 

16-1 

29.4 

46-5 

67.6 

92-6 

121*5 

53 

1.5 

7.0 

i6.3 

29.6 

46-8 

68.0 

93.0 

I22*O 

54 

1.6 

7.1 

i6-5 

29  9 

47-i 

68-3 

93-5 

122-5 

55 

1.6 

7.2 

16.7 

3o.i 

47  5- 

68.7 

93.9 

123-1 

56 

1.7 

7.3 

«'••« 

3o-4 

47-8 

69.1 

94.4 

123-6 

57 

1.8 

7-5 

17.1 

3o-6 

48-i 

69.5 

94-8 

124*1 

58 

i-8 

7.6 

17-3 

3o«9 

48-4 

69.9 

95-3 

124-6 

59 

1.9 

7-7 

17-5 

3i.i 

48-8 

7o.3 

95.7 

125.  I 

442 


SPHERICAL    ASTRONOMY. 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
2  sin'  J  P 

~~ 


Sec. 

gm 

9m 

10m 

llm 

12m 

13m 

14m 

ii 

H 

u 

H 

u 

ti 

o 

125-7 

159.  o 

196-3 

237-5 

282-7 

33i.8 

384-7 

i 

126*2 

169.6 

197-0 

238-3 

283-5 

332-6 

385.6 

2 

126-7 

160-2 

197.6 

239-0 

284-2 

333-4 

386-6 

3 

127-2 

160-8 

198-3 

239.7 

285-0 

334-3 

387-5 

4 

127-8 

161-4 

198.9 

240-  4 

285-8 

335-2 

388-4 

5 

128-3 

162-0 

199-6 

24l-2 

286-6 

336-0 

389.3 

6 

128-8 

162-6 

200«  3 

241-9 

287-4 

336-9 

390-2 

7 

129.3 

[63-2 

200-9 

242-6 

288-2 

337.7 

391  .1 

8 

129.9 

i63-8 

201-6 

243.3 

289-0 

338-6 

392.1 

9 

i3o-4 

164-4 

2O2-2 

244-1 

289-8 

339.4 

393-o 

10 

i3i  «o 

i65-o 

202.9 

244-8 

290-6 

340-3 

393.9 

ii 

i3r.5 

i65.6 

203-6 

245.5 

291.4 

34i-2 

394-8 

12 

l32-O 

166-2 

2O4'2 

246-3 

292-2 

342-0 

395-8 

i3 

i32.6 

166-8 

204-9 

247.0 

293-0 

342.9 

396.7 

i4 

i33-i 

167-4 

2o5-6 

247-7 

293-  8 

343.7 

397.6 

i5 

i33.6 

168-0 

206.3 

248-5 

294-6 

344-6 

398-6 

16 

i34-  2 

168-6 

206-  9 

249-2 

295-4 

345.5 

399-5 

17 

i34-7 

169-2 

207.6 

249-9 

296-2 

346-4 

4oo-5 

18 

i35«3 

169-8 

208-3 

25o-7 

297-0 

347-2 

4oi-4 

*9 

i35-8 

170.4 

208.9 

25i-4 

297-8 

348-1 

402.3 

20 

i36.3 

171*0 

209.6 

252-2 

298-6 

349-0 

4o3-3 

21 

i36-9 

171.6 

210-3 

253.0 

299-4 

349-8 

4o4«a 

22 

i37-4 

172.2 

2II-O 

253-6 

300-2 

35o-7 

4o5.i 

23 

i38-o 

172.9 

211  -7 

254-4 

3oi  .0 

35i-6 

4o6«o 

24 

i38-5 

i73-5 

212-3 

255-1 

3oi.8 

352.5 

407.0 

25 

i39«i 

174'  i 

2l3-O 

255-9 

3o2-6 

353-3 

4o8-o 

26 

i39-6 

174      7 

2l3.7 

256-6 

3o3-5 

354-2 

408-9 

27 

i4o-2 

i75-3 

2i4-4 

257-4 

3o4-3 

355-1 

409.9 

28 

i4o«7 

175.9 

2l5-I 

258-1 

3o5-i 

356-0 

4io.8 

a9 

i4i-3 

176-6 

2i5-8 

208-9 

3o5-9 

356-9 

4n»7 

TABLES. 


443 


TABLE  V. — (Continued.) 
For  the  Reduction  to  the  Meridian  :  showing  the  value  of 


A  = 


_  2  sin2  1  P 


sin  1' 


Sec. 

gm 

9m 

10m 

12m 

13m 

14m 

3o 

i4i-8 

^77-2 

216.4 

259-6 

3o6-7 

357.7 

4l2-7 

3i 

142-4 

177.8 

217-1 

260-4 

307-5 

358-6 

4i3.6 

32 

i43.o 

178-4 

217-8 

261  -i 

3o8-4 

359.5 

4i4-6 

33 

i43.5 

179-0 

218-5 

261  -9 

309-2 

36o-4 

4i5-5 

34 

i44-i 

179-7 

219-2 

262-6 

3io-o 

36i.3 

4i6-5 

35 

i44-6 

i8o.3 

219.9 

263-4 

3io-8 

362-2 

4r7-5 

36 

i45.2 

180-9 

22O-6 

264-1 

3n-6 

363  .  i 

418-4 

37 

i45«8 

181-6 

221-3 

264-9 

3i2-5 

364-0 

419.4 

38 

i46-3 

182-2 

222«  0 

265-7 

3i3.3 

364-8 

420.3 

39 

146-9 

182-8 

222-7 

266-4 

3i4-i 

365-7 

421.3 

4o 

i47-5 

i83-5 

223-4 

267-2 

3i5-o 

366-6 

422-2 

4i 

i48«o 

184-1 

224-1 

267.9 

3i5-8 

367-5 

423-2 

42 

i48-6 

184-7 

224-8 

268.7 

3i6.6 

368-4 

424-2 

43 

149-2 

i85-4 

225-5 

269-5 

3i7-4 

369-3 

425.1 

44 

149-7 

186-0 

226-2 

270-3 

3i8-3 

370*2 

426-1 

45 

i5o-3 

186-6 

226-9 

271  -o 

3i9.i 

37i-i 

427-0 

46 

1  5o  •  9 

187-3 

227-6 

271-8 

319-9 

372-0 

428-0 

47 

i5i«5 

187.9 

228-3 

272-6 

320-8 

372-9 

429.0 

48 

152.0 

i88.5 

229-0 

273.3 

3ai.6 

373-8 

429.9 

49 

i52.6 

189.2 

229.7 

274-1 

322.4 

374-7 

43o.9 

5o 

i53.2 

189-8 

23o-4 

274.9 

323-3 

375.6 

43i-9 

5r 

i53-8 

.  190*5 

a3i.«i 

275.6 

324-1 

376.5 

432.8 

52 

154.4 

191-1 

23i-8 

276-4 

325-0 

377-4 

433-8 

53 

i54-9 

191-8 

232.-S 

277-2 

325-8 

378.3 

434-8 

54 

i55-5 

192.4 

233-2 

278-0 

326-7 

379-3 

435-8 

55 

i56-i 

193.1 

234-0 

278-8 

327-5 

38o-2 

436.7 

56 

i56-7 

193-7 

234-7. 

279-5 

328-4 

38i-i 

437-7 

57 

i57-3 

194-4 

235-4 

280-3 

329.2 

382-0 

438.7 

58 

157.8 

195-0 

236-1 

281-1 

33o.o 

382.9 

439-7 

59 

i58-4 

195-7 

236.8 

281-9 

33o.9 

383-8 

44o-6 

444 


SPHERICAL    ASTRONOMY. 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
2  sin2 1  P 
"ihTT7" 


A  = 


Sec. 

15ra 

16m 

I7m 

18m 

19m 

20m 

21m 

o 

44i-6 

502-5 

567-2 

635-9 

708-4 

784-9 


865-3 

i 

442-6 

5o3.5 

568-3 

637.o 

709-7 

786-2 

866-6 

2 

443-6 

5o4-6 

569.4 

638-2 

710.9 

787-5 

868-0 

3 

444-6 

5o5-6 

57o-5 

639-4 

712-1 

788-8 

869.4 

4 

445-6 

5o6.7 

571-6 

64o-6 

7i3-4 

790-1 

870.8 

5 

446-5 

5o7.7 

572-8 

64r-7 

714-6 

791.4 

872-1 

6 

447-5 

5o8-8 

573.9 

642-9 

715.9 

792-7 

873-5 

7 

448-5 

5o9.8 

575-0 

644-i 

717-1 

794-0 

874-9 

8 

449.5 

510.9 

576.1 

645-3 

718-4 

795.4 

876-3 

9 

45o.5 

5ii.9 

577-2 

646-5 

710-6 

796-7 

877-6 

10 

45i-5 

5i3.o 

578-4 

647-7 

720.  9 

798-o 

879-0 

ii 

452.5 

5i4-o 

579-5 

648.9 

722-1 

799-3 

880-4 

12 

453-5 

5i5.i 

58o-6 

65o-o 

723-4 

800.7 

881-8 

i3 

454-5 

5i6-i 

58i.7 

65i-2 

724-6 

802-0 

883-2 

i4 

455-5 

5i7-2 

582.9 

652-4 

725-  9 

8o3-3 

88^-6 

i5 

456.5 

5i8-3 

584-o 

653-6 

727-2 

8o4-6 

886.0 

16 

457.5 

5i9-3 

585-1 

654-8 

728-4 

8ob-o 

887-4 

17 

458-5 

520-4 

586-2 

656-0 

729-7 

807-3 

888-8 

18 

459-5 

521-5 

587.4 

657-2 

73o-9 

808-6 

890-2 

'9 

46o.5 

522-5 

588-5 

658-4 

732-2 

809-9 

891-6 

20 

46i-5 

523-6 

589-6 

659-6 

733-5 

8il.3 

893-0 

21 

462-5 

524-6 

590-8 

'  660-8 

734-7 

812-6 

894-4 

22 

463.5 

525-7 

591-9 

662-0 

736-o 

8i3-9 

895-  8 

23 

464-5 

526-8 

593-o 

663-2 

737-3 

8i5-2 

897.2 

24 

465-5 

527-9 

594-2 

664-4 

738.5 

816-6 

808-6 

25 

466-5 

528-9 

595-3 

665-6 

739-8 

817.9 

9oo-o 

26 

467-5 

53o-o 

596-5 

666-8 

74i-i 

819-2 

001-4 

27 

468-5 

53r-i 

597-6 

663-0 

742-3 

820-5 

902«8 

28 

469-5 

532-2 

598-7 

669.2 

743-6 

821.9 

9o4-2 

29 

470-5 

533-2 

599-9 

670-4 

744-  9 

823-2 

.9o5-ft 

TABLES 


445 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
2  sin2  i  P 

A    __    ^ 

sin  I7'     ' 


Sec. 

15m 

16m 

I7m 

18m 

19m 

20» 

21m 

3o 

471-5 

534-3 

601  -o 

671-6 

746.2 

824  6 

907-0 

3i 

472  6 

535-4 

602  •  2 

672-8 

747-4 

825.9 

908-4 

32 

473-6 

536-5 

6o3-3 

674-1 

748-7 

827.3 

909-8 

33 

474-6 

537.6 

6o4-5 

675-3 

75o«o 

828-6 

911-2 

34 

475-6 

538-7 

6o5-6 

676-5 

75i-3 

829.9 

912-6 

35 

476-6 

539-7 

606-8 

677-7 

752-6 

83i-2 

914-0 

36 

477-6 

54o-8 

607.9 

678.9 

753-8 

832-6 

9r5-5 

37 

478-7 

54r-9 

609.  r 

680.  i 

755-  1 

833-9 

916.9 

38 

479-7 

543-0 

610-2 

681.3 

756-4 

835-3 

918-3 

39 

480-7 

544-i 

611.4 

682-6 

757-7 

836-6 

919.7 

4o 

48i-7 

545-2 

612.5 

683-8 

759-0 

838-0 

921.1 

4r 

482-8 

546-3 

6i3.7 

685-0 

760-2 

839-3 

922.5 

42 

483-8 

547-4 

614-8 

686-2 

761-5 

84o-7 

923.9 

43 

484-8 

548-4 

616-0 

687-4 

762-8 

842-0 

925-3 

44 

485-8 

549-5 

617-2 

688-7 

764-1 

843-4 

926.8 

45 

486-9 

55o-6 

6i8-3 

689-9 

765-4 

844-7 

928-2 

46 

487-9 

55i-7 

619-5 

691  -  r 

766.7 

846-1 

929-6 

47 

488-9 

552-8 

620-6 

692.4 

768-0 

847-5 

93i  «o 

48 

490-0 

553-9 

621-8 

693-6 

769-3 

848-9 

932.4 

49 

491-0 

555-0 

623-0 

694-  8 

770-6 

85o-2 

933.8 

5o 

492-0 

556-1 

624.1 

696-0 

771-9 

&5i-6 

935.2 

5i 

493.1 

557-2 

625-3 

697.3 

773-i 

852.9 

936-6 

52 

494-r 

558-3 

626-5 

698-5 

774-5 

854-3 

938.i 

53 

495    2 

559.4 

627-6 

699.7 

775-8 

855-7 

939.5 

54 

496.2 

56o-5 

628-8 

701  -o 

777.1 

857-1 

94o  "Q 

55 

497-2 

56i-6 

63o-o 

7O2    2 

778-4 

858-4 

942.3 

56 

498.3 

562  7 

63r-2 

703^5 

779-7 

859-8 

943.8 

57 

499-3 

563-9 

632-3 

704  '7 

781-0 

861-1 

945.2 

58 

5oo-3 

565-0 

633-5 

705-7 

782.3 

862-5 

946-6 

59 

5oi-4 

566-x 

634-7 

707- 

783-6 

863-9 

948-1 

SPHERICAL    ASTRONOMY. 


TABLE  V. — (Continued.) 
For  the  Reduction  to  the  Meridian  :  showing  the  value  of 

A  —  2  siQ2  i  P 
~      sin  l/r~~' 


Sec. 

22m 

23m 

24m 

25m 

26™ 

27m 

28m 

0 

949-6 

io37-8 

1129-9 

1225.9 

i325-9 

1429-7 

i537-5 

i 

96  !•  o 

1039.3 

n3[-4 

1227.5 

i327-6 

i43i-4 

i539-3 

2 

962.4 

io4o«8 

n33-o 

1229-2 

i329.3 

i433.2 

i54i-f 

3 

953.8 

io42«3 

n34-6 

i23o-8 

i33i-o 

i434-9 

1542-9 

4 

955.3 

io43-8 

u36-2 

1232-5 

i332.7 

i436-7 

i544-8 

5 

956.7 

io45-3 

1187.8 

1234.1 

i334-4 

1438-5 

i546-6 

6 

968.2 

io46-8 

ii39-3 

1235-7 

i336-i 

i44o.3 

i548-4 

7 

959.6 

io48-3 

ii4o«9 

i237-3 

i337-8 

1442-1 

i55o-2 

8 

961  >  i 

1049-8 

ii42-5 

1239-0 

i339-5 

1443-9 

1  552-  1 

9 

962.5 

io5i  -3 

n44'0 

1240-  6 

i34i-2 

i445-6 

:553-9 

10 

963.9 

io52-8 

ii45-6 

1242-3 

1342-9 

1447.4 

r555:8 

ii 

965-4 

io54-3 

1147-2 

1243-9 

i344-6 

1449.2 

i557-6 

12 

966.9 

io55-9 

ii48-8 

1245-6 

i346-3 

i45i  «o 

i559-5 

i3 

968-3 

1067.4 

n5o-4 

1247-2 

i348-o 

i452.8 

i56i-3 

14 

969.8 

io58-9 

Il52-0 

1248-9 

1349-7 

i454-5 

i563-2 

i5 

971-2 

1060-4 

ii53-6 

i25o-5 

i35r-4 

i456-3 

i565-o 

16 

972-7 

1062-0 

u55-2 

1252-2 

i353-2 

i458-i 

i566.9 

i? 

974-1 

io63-5 

n56-8 

1253-8 

1354-9 

1459.9 

i568-7 

18 

975.5 

io65-o 

n58-3 

1255-5 

1356-6 

i46i-6 

i57o.5 

'9 

977.0 

1066-  5 

1159-9 

1257-1 

i358-3 

i463-4 

1572-4 

20 

978.5 

1068-  i 

»6l.5 

T258.8 

i36o-i 

i465-2 

i574-3 

21 

979.9 

1069-6 

ii63-i 

1260.4 

i36i-8 

1466-9 

1576-1 

22 

981-4 

1071-1 

1164-7 

1262-1 

i363-5 

i468-7 

1578-0 

23 

982.9 

1072-6 

ii66-3 

1263-7 

i365.2 

i47<>-5 

i579.8 

24 

984-4 

1074-2 

1167-9 

1265-4 

i  367-0 

1472-3 

i58i.7 

25 

985-8 

1075-7 

1169-5 

i  267  •  o 

i368-7 

i474-o 

1583-5 

26 

987-3 

1077-2 

1171-1 

1268-7 

1370-4 

1475-9 

i585-3 

27 

988-8 

1078-7 

1172-7 

1270-  3 

1372-1 

i477'7 

1587-2 

28 

990.3 

io8o-3 

u74-3 

1272- 

i373-9 

i479'  5 

1589.1 

29 

991-8 

1081-8 

1175.9 

1273.7 

i375-6 

i48r-3 

1590-9 

i 

i 

TABLES. 


447 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
_  2  sin2  -*  P 
~      sin  1"    ' 


Sec. 

22m 

23m 

24m 

25m 

26m 

27" 

28*  . 

3o 

993-2 

io83-3 

1177.5 

1275.4 

i377-4 

i483-i 

1592-7 

3i 

994-7 

1084-8 

1179.1 

1277.1 

i379.o 

1484-9 

1594-6 

32 

996-2 

1086-4 

1180.7 

1278-8 

i38o-8 

i486-7 

1596-5 

33 

997-6 

1087-9 

1182.3 

1280.4 

i382-5 

i488.5 

1598-3 

34 

999.1 

1089.5 

ii83-9 

1282-1 

1384-2 

1490-3 

1600-2 

35 

1000-6 

1091  -o 

n85-5 

1283-8 

1385-9 

1492.1 

1602-  i 

36 

i  002-  i 

1092-6 

1187-1 

1285-5 

i387-7 

1493.9 

i  6o4  •  o 

37 

ioo3-5 

1094.1 

1188-7 

1287-1 

1389-4 

1495-7 

1605-9 

38 

ioo5-o 

1095.7 

1190-3 

1288-8 

1391-2 

i497-5 

1607.7 

39 

i  006  •  5 

1097-2 

1191.9 

i  290  •  5 

i392.9 

1499-3 

1609.6 

4o 

1008-0 

1098-8 

1193.5 

1292-2 

i394-7 

i5oi  -i 

i6n.5 

4i 

1009.4 

1100.3 

1195.1 

1293-8 

i396-4 

i5o2-9 

i6i3-3 

42 

1010.9 

1101-9 

1196.7 

1295-  5 

i398-2 

i5o4-7 

i6i5-a 

43 

IOI2-4 

1*53.4 

1198-3 

1297.2 

1399.9 

.1506-5 

1617-1 

44 

1013.9 

iio5«o 

1199.9 

1298-9 

i4oi«7 

i5o8-4 

1619-0 

45 

ioi5«4 

iio6-5 

I2or  «5 

i3oo«5 

i4o3-4 

l5[O-2 

1620-8 

46 

1016-9 

1108-1 

I2O3«I 

1302.2 

i4o5-2 

l5l2«O 

1622-7 

47 

1018.4 

1109-6 

1204-7 

i3o3-9 

1406.9 

i5i3-8 

1624-6 

43 

1019-9 

1III.  2 

1206.4 

i3o5-6 

1408-7 

i5i5.6 

1626.5 

49 

1021  -4 

1112-7 

I2O8«O 

i3o7-3 

i4io«4 

i5i7.4 

1628-3 

5o 

I022-  & 

ni4-3 

1209.6 

1  309  «  o 

l4l2«2 

1519.2 

i63o-2 

5i 

io?4-3 

ui5-8 

I2II.2 

i3io«7 

i4i3'9 

l52I  -O 

i632-i 

52 

1025-8 

1117-4 

1212*9 

i3i2-4 

i4]5-7 

l522-9 

1634.0 

53 

1027.3 

1118-9 

i2i4-5 

i3i4-r 

i4i7-4 

1524-7 

i635-9 

54 

1028-8 

1I2O-5 

1216-  i 

i3i5-7 

1419-2 

i526-5 

i637-7 

55 

io3o«3 

fI22-O 

1217.7 

i3i7-4 

1420-9 

i528-3 

i639-6 

56 

1:531-8 

H23-6 

1219.4 

1319-1 

1422.7 

i53o-2 

1  64  1  •  5 

57 

io33-3 

II25-I 

1221  -0 

i  320-  8 

1424-4 

i532-o 

1643-3 

58 

io34-8 

1126-7 

1222.6 

1322.5 

1426-2 

i533-8 

1645-2 

59 

io36-3 

1128-3 

1224-2 

1324-2 

1427-9 

i535-6   1647-1 

448 


SPHERICAL    ASTRONOMY. 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
_  2  sin2  i  P 
~~      sin  1"    * 


Sec. 

29m 

30m 

31m 

32ra 

33m 

34m 

35m 

0 

1649-0 

I764r6 

i884-o 

2007-4 

2i34-6 

2266-6 

2400-6 

i 

i65o«9 

1766-6 

1886-0 

2009.4 

2i36.8 

2267-8 

2402-9 

2 

1662.8 

1768-5 

1888-  o 

2OI1  -5 

9  i  38.  9 

2270-0 

24o5-2 

3 

1664.7 

1770-5 

1890-0 

2oi3-6 

2l4l  •  I 

2272-  2 

2407  -  5 

4 

1666-6 

1772-4 

1892-1 

2016.7 

2143.2 

2274-5 

2409-8 

5 

1658-5 

1774-4 

1894-1 

2017-8 

2145-3 

2276-7 

2412-0 

6 

1660-4 

1776-3 

1896-1 

2019-9 

2i47-5 

2278-9 

24i4-3 

7 

1662-3 

1778-3 

1898-1 

2022-0 

2149-7 

2281.2 

2416-  6 

8 

1664-2 

1780-3 

I  900  •  2 

2024-  I 

2161-8 

2283-4 

2418-9 

9 

1666-1 

1782-3 

I9O2.2 

2026-2 

2153-9 

2285-6 

2421  -2 

10 

1668-0 

1784-2 

1904.3 

2028-3 

2  i  56-  i 

2287-8 

2423-5 

ii 

1669-9 

1786-2 

1906.3 

2o3o-5 

2i58-3 

2290-0 

2425.8 

12 

1671-9 

1788-2 

1908-4 

2032-5 

2i6o-5 

2292-3 

2428.1 

i3 

i673-  8 

1790-1 

I9I0.4 

2o34-6 

2162-6 

2294.5 

243o-4 

i4 

i675-7 

1792-1 

19I2.4 

2o36«7 

2164.  8 

2296-8 

2432-7 

i5 

1677-6 

1794-1 

I9I4-4 

2o38.8 

2166-9 

2299-0 

2435-0 

16 

1679.5 

1796-1 

1916-6 

2040  «  9 

2  1  69  -  I 

230I  «3 

2437-3 

'7 

1681-4 

1798-1 

1918.5 

2o43-  o 

2I7f.2 

23o3-6 

2439-6 

8 

i683-3 

1800.  o 

1920-6 

2045  •  i 

2173.4 

23o5-8 

2441-9 

'9 

i685-2 

1802-0 

1922.6 

2047  '  2 

2176-6 

23o8-o 

2444-2 

20 

1687-2 

i8o4-o 

I924-7 

2049.3 

2177-8 

23lO-2 

2446-5 

21 

1689-1 

i8o5-9 

1926-7 

2o5i  -4 

2179-9 

2312-4 

2448-8 

22 

1691  -o 

1807-9 

1928-8 

2053-5 

2I82.I 

23i4-7 

2461-1 

23 

1692-9 

1809.9 

1930.8 

2055-7 

2i84-3 

2316-9 

2453-4 

24 

1694-8 

1811  -9 

i932-9 

2067-8 

2186-6 

2319-2 

2455.7 

25 

1696-7 

1813.9 

1935-0 

2069-9 

2188.6 

2321-5 

2458-0 

26 

1698-6 

i8i5-8 

1937-0 

2062-0 

2190.8 

2323-7 

2460.3 

27 

1700-6 

1817-8 

1939-0 

2064  •  i 

2193.0 

2325-9 

2462-6 

28 

1702-6 

1819-8 

1941-1 

2066-2 

2196.2 

2328-2 

2464-9 

29 

1704-4 

1821-8 

i  943  .  i 

2068.3 

2197-3 

2467.2 

TABLES. 


449 


TABLE  V.— (Continued.) 

For  the  Reduction  to  the  Meridian  :  showing  the  value  of 
2  sin2  I  P 


Sec. 

29m 

30m 

31m 

32m 

33m 

34m 

35m 

3o 

1706-3 

1823-8 

1945.2 

2070-4 

2199.5 

2332.7 

2469.5 

3i 

1  708  •  2 

1825.8 

1947.2 

2072-6 

22OI  «7 

2334-9 

2471-8 

32 

I7I0.2 

1827-8 

1949.3 

2074-7 

2203.9 

2337-2 

2474.2 

33 

I7I2.I 

1829-8 

i95i.3 

2076-8 

2206.1 

2339.4 

2476-5 

34 

I7l4'0 

i83i-8 

1953.4 

2078*9 

2208.3 

234r-7 

2478-8 

35 

I7I5.9 

1833-8 

i955-  5 

2081  -o 

221O-5 

2343-9 

2481-1 

36 

1717.9 

i835-8 

1957-6 

2083-2 

2212-7 

2346-2 

2483-5 

3? 

1719-8 

i837-8 

1959-6 

2085-3 

2214-9 

2348-5 

2485-8 

38 

I72I.7 

1839-8 

1961-7 

2087.4 

2217-1 

235o.7 

2488.1 

39 

1723.6 

i84i-8 

i963-7 

2089-6 

2219-3 

2353-0 

2490.4 

4r 

1725.6 

i843-8 

I965.8 

2091-7 

2221  -5 

2355-2 

2492-8 

4: 

1727.5 

1845-8 

1967-8 

2093-8 

2223-7 

2357-5 

2495-1 

42 

1729.5 

1847-  8 

1969.9 

2095-9 

2225*9 

2359-7 

2497.4 

43 

I73I.5 

1849-8 

1972-0 

2098-0 

2228-1 

236i  -9 

2499-7 

44 

1733.4 

i85i-8 

1974-1 

2IOO-2 

2230-3 

2364-a 

25O2«I 

45 

1735-3 

1853-8 

1976-1 

2102-3 

2232-5 

2366-4 

25o4«4 

46 

I737.-J 

1855-8 

1978-2 

2io4«5 

2234-7 

2368-7 

25o6.7 

4? 

1739.2 

1857.8 

1980-3 

2106.  6 

2236-9 

2371-0 

2509.  o 

48 

I74l-2 

1859.8 

1982-4 

2108-8 

2239-1 

2373.3 

25ri-4 

49 

1  743-.  I 

1861-8 

1984-4 

2IIO«9 

2241  -3 

2375.5 

25t3.7 

5o 

1745.1 

1863-8 

1986-5 

2Il3«  I 

2243-5 

2377  -8 

25i6.i 

5: 

1747-0 

i865.8 

1988-6 

2Il5«2 

2245-7 

238o-i 

25i.8-4 

52 

1749-0 

1867-8 

1990-7 

2117.4 

2247-9 

2382-4 

2520-8 

53 

1750.9 

1869.8 

1992.7 

2119.6 

225O-I 

2384  6 

2523-1- 

54 

1752-9 

1871-8 

1994-8 

2121.7 

2252-3 

2386-9 

2525-4 

55 

1754-  8 

i873.  8 

1996-9 

2123-8 

2254-5 

2389.2 

2527.7- 

56 

i756-8 

1875-9 

1999.0 

2126-0 

2256-7 

239i-5 

253o-i 

57 

i758.7 

1877-9 

2001  -0 

2  I  28  .  I 

2258.9 

2393-7 

2532.4 

58 

1760-7 

1879.9 

2OO3-I 

2i3o-3 

2261  •  i 

23o6-o 

2534.8 

59 

1762.6 

1882-0 

2005-3 

2132-4 

2263-4 

2398-3 

2537-1 

i 

450 


SPHERICAL    ASTRONOMY. 


TABLE  VI. 

For  the  second  part  of  the  Reduction  to  the  Meridian :  showing  the  value  of 


2sin4-*P 
sin  1"    * 


Minutes 

0s 

10« 

20s 

30' 

40" 

50s 

M 

M 

a 

M 

M 

// 

5 

O-OI 

o.oi 

O-OI 

O-OI 

O«OI 

O-OI 

6 

O«OI 

O«OI 

O-OI 

0.02 

0-02 

O«O2 

7 

O«O2 

O«O2 

o«o3 

o«o3 

o-o3 

0-04 

8 

0-04 

o«o4 

o-o5 

o«o5 

o-o5 

o«o6 

9 

o«o6 

0*07 

0-08 

0-08 

0-08 

0-09 

10 

0-09 

O»IO 

O«II 

O«I  I 

O«I2 

o-i3 

ii 

o.i4 

o-i5 

o.i5 

o«i6 

0-17 

0-18 

12 

0-19 

O-2O 

O-22 

O.23 

0.24 

0-25 

i3 

0-27 

0.28 

o«3o 

o-3i 

0-33 

0-34 

i4 

0-36 

o-38 

0.39 

o-4i 

0-43 

o-45 

i5 

0-47 

o«49 

0-52 

0-54 

o-56 

o.59 

16 

O'6i 

o-64 

0-67 

0*69 

0-72 

o.75 

i? 

0.78 

0-81 

o-84 

0-88 

0-91 

0-95 

18 

0-98 

I  «O2 

i  «o6 

1.09 

i-i3 

1.18 

'9 

I  .22 

1-26 

i-3o 

1-35 

i«4o 

i-44 

20 

1.49 

1-54 

i  «6o 

i.65 

1.70 

1.76 

21 

1.82 

1.87 

i-93 

1.99 

2.06 

2«I2 

22 

2.19 

2-25 

2-32 

2.39 

2.46 

2-54 

23 

2«6l 

2.69 

2.77 

2-85 

2.93 

3.oi 

24 

3.10 

3.18 

3.27 

3-36 

3.45 

3.55 

25 

3-64 

3.74 

3-84 

3.94 

4-o5 

4-i5 

26 

4.26 

4-37 

4-48 

4-6o 

4.72 

4-83 

27 

4.96 

5-o8 

5  -20 

5-33 

'5-46 

5-6o 

28 

5.73 

5-87 

6«oi 

6-i5 

6-3o 

6-44 

29 

6.59 

6-75 

6*90 

7.06 

7.22 

7-38 

3o 

7-55 

7.72 

7.89 

8-06 

8-24 

8.42 

3i 

8-61 

8-79 

8-98 

9.17 

9.37 

9-57 

32 

9.77 

9.97 

10.  18 

10.39 

io«6i 

10-82 

33 

ii  -o4 

ii  .27 

ii  «5o 

ii.  73 

ii  .96 

I  2«  2O 

34 

12-44 

12*69 

12-94 

13-19 

i3-45 

i3.7i 

35 

13.97 

!4-24 

i4-5i 

14-78 

i5-o6 

i5-15 

TK1GONOMETKICAL  FOUMULJE. 


I.  Equivalent  expressions  for  sin  & 


1.     cos  x  .  tan  x. 
cos  x 


2. 


cot  x 


7. 
8. 
9. 

n 

A/l  —  cos  2  * 

• 

2 

2  tan  £  x 

1  +  tan1  Jar" 
2 

cot  \  x  +  tan  J  i 
sin  (30°  +  a;)  - 

B 

sin  (30°  —  x) 

13. 


VI 

11.  2  sin«(45°  -f  i  *)  —  1. 

12.  1  —  2  sin8  (45°  —  £ar). 

1  -  tan2  (45°  -  ^  ar) 
1  4-  tan2  (45°  -  £  *)  * 

tan  (45°  +  j  *)  -  tan  (45°  -  $  a?) 
tan  (45°  +  i  *)  +  tan  (45°  —  J  ar)' 

16.     sin  (60°  +  a;)  —  sin  (60°  —  x). 

1 


16. 


oowcant  x 


452 


SPHERICAL    ASTRONOMY. 


D.  Equivalent  expressions  tor  cos 


1. 

sin  x 
tan  x 

2. 
3. 

sin  x  .  cot  x. 

Vl  —  sin8  x. 

4. 

1 

Vl  4-  tan2  x 

5. 

cot  x 

Vl  4-  cot8"* 

0. 

cos2  ^  a;  —  sin8  £  as. 

7. 

1  —  2  sin8  Jar. 

8. 
9. 

2  cos'  J  x  -  1. 

4/l  -f-  cos  2  a: 

2 

10. 

1  —  tan2  J  x 

1  H   tan2  J*' 

11. 

cot  J  ar  —  tan  J  x 

cot  J  a;  4-  tan  J  a:  ' 

12. 

1 

1  4"  tan  x  .  tan  ^  a?  * 

13. 

2 

tan  (45°  4-  }  *)  4-  cot  (45°  4-  j  a?) 

14. 

2  cos  (45°  4-  J  x)  cos  (45°  —  Jar). 

15. 

cos  (60°  4-  a:)  4-  cos  (60°  —  *). 

16. 

1 
secant  x' 

TRIGONOMETRICAL   FORMULAE 


453 


HI.  Equivalent  expressions  for  tan  x. 


1. 


sm  x 

COB  X 

1 

cot  a:' 


4. 


sm  x 


6. 


7. 


8. 


Vl  —  cos8  x 

COS  X 

2  tan  £  a; 
1  —  tan*  Jar" 

2  cot  j  x 
cot8  £  a;  —  1 ' 

2 


cot  J  *  —  tan  f  x 
9.    cot  x  —  2  cot  2  *. 
1  —  cos  2  x 


10. 


11. 


sin  2  x 

sin  2  # 
+  cos  2  or* 


19. 


—  cos  2  a; 
1  4-  cos  2  a? 

tan  (45°  +  £  *)  —  tan  (45' 


4.54  SPHERICAL    ASTRONOMY. 

IV.  Kelative  to  two  arcs  A  and  B. 

1.  sin  (A  4-  B)       =  sin  A  .  cos  B  +  cos  A  .  sin  B. 

2.  sin  (A  —  B)       =  sin  A  .  cos  2?  —  cos  A  .  sin  ./?. 

3.  cos  (A  -\-  B)       =  cos  A  .  cos  J?  —  sin  A  .  sin  J5. 

4.  cos  (^4  —  -B)       =  cos  A  .  cos  B  +  sin  -4  .  sin  B. 
5  tanU  +  JS)       _     tan  ^  + tan  ^ 


tan  A  —  tan  J5 
6.     tan   ^-5        =     ---- 


7.     sin  (45°  ±5) 


1  ±  tan 
9.     tan  (45-  ±1?)    ^H 


sin  B 
10. 


1  db  sin  B  cos  .5 

11. 


sin  (-4  +  B)  tan  -4  +  tan  B  '__  cot  ^  +  cot  A 

12'    sin  (A  —  ^?)  =  tan  A  —  tan  £  ~~  cot  B  -  cot  ^4 

cos  (A  +  B)  cot  J?  —  tan  A  _  cot  ^4  —  tan  ^ 

13'     cos  (^4  —  B)  ~  cot  B  -f  tan  -4  ~~  cot  A  +  tan  B  ' 

sin  ^.  +  sin  ^         tan  ^  (^1  +  B) 
14. 


15. 


sin  A  —  sin  £         tan  %  (A  —  B) 

cos  jg-f  cos^L        cot  |(^4  +  ^g) 
cos  ^  —  cos  J.    "~  tan  i  '^1  —  B) ' 

[continued. 


TRIGONOMETRICAL  FORMULAE.  455 

IV.  continued.     Eelative  to  two  arcs  A  and  B. 

16.  sin  A  .  cos  B  =  £  sin  (A  +  B)  +  £  sin  (A  —  B). 

17.  cos  A  .  sin  B  =  $  sin  (A  +  B)  —  %  sin  (A  —  B). 

18.  sin  A  .  sin  B  =  %  co*  (A  —  B)  —  %  cos  (A  +  ^?). 

19.  cos  A  .  cos  5  =  J  cos  (J.  -f  B)  +  i  cos  (Ji  —.  B). 

20.  sin  J.  +  sin  £  =  2  sin  i  (^4  +  .5)  .  cos  %  (A  —  B). 

21.  cos  A  H-  cos  ^  =  2  cos  J  (^4  +  B)  .  cos  ^  (A  —  ,5).   , 

sin  (A  +  -B) 


sin  (  A  +  £) 
23.     cot  A  +  cot  ^      =—-—. 


24.  sin  ^1  -  sin  B      =  2  sin  1  (^  -  ^)  .  cos  %  (A  +  B). 

25.  cos  B  —  cos  A      =  2  sin  ^  (A  —  B)  .  sin  i  (^  +  •#) 

sin  (^  -  B) 


26. 


sin  (A  - 

27.    cot  B  —  cot  A      =  - 


sin  -4  .  sin  B ' 

28.  sin2  A  —  sin2  5  ) 

V  =  sin  (.4  —  B)  .  sin  (^4  +  B). 

29.  cos2  J?  —  cos2  A  ) ' 

30.  cos1  A  —  sin2  B    =  cos  (^4  —  B)  .  cos  (^4  +  B). 

sin  (A  —  £)  .  sin  (A  +  B) 

31.  tan'  4  -  trf .»  -  -     ^.^A- 

f  sin  (A  —  B)  .  sin  (A  +  -5) 


456  SPHERICAL   ASTRONOMY. 

V.  Differences  of  trigonometrical  lines. 

1.  A  sin  a;     =  -f  2  sin  -J  A  x  .  cos  (a;  -f  i  A  a?). 

2.  A  cos  a;     =  —  2  sin  \  A  x  .  sin  (x  +  %  A  ar). 

3.  A  tan  a:    =  +  sin  A  * 


4.     A  cot  a;    = 


cos  x  .  cos  (x  +  A 
sin  A  # 


sin  ar  .  sin  (x  +  A  a;)  ' 

5.  A  sin2  x    =  +  sin  A  a;  .  sin  (2  x  +  A  x). 

6.  A  cos2  x   =  —  sin  A  a:  .  sin  (2  #  -f  A  x}. 

7.  A  tan2  x   =  +  Sin  A  *  •  8iD  (2  *  +  A  ^) 

cos2  x  .  cos2  (a;  -f-  A  a:) 


8.     A  cot°  x  =  - 


sin2  x  .  sin2  (a?  -f  A  a?) 


VI.  Differentials  of  trigonometrical  lines. 

1.  d  sin  x      =.  -f-  d  x  .  cos  x. 

2.  d  cos  a;     rr  —  d  x  .  sin  x. 

da? 


8.     d  tan  a;     = 


cos*  x ' 


dx 

4.  d  cot  x     = — -  . 

siir  x 

5.  d  sin2  x    =  +  2  d  a;  .  sin  x  .  cos  ?. 

6.  d  cos8  x    =  —  2  d  a;  .  sin  x  .  cos  jr. 

7.  d  tan'*    =  +2d*-tan*. 

cos2  a: 


Q  j          j  g  6  vJ  •C     •     C\ 

sin2  x 


TRIGONOMETRICAL   FORMULAE.  457 


VII.   General  analytical  expressions  for  the  sides  end  angles 
of  any  spherical  triangle. 


1.  cos  S    =  cos  .4    .  sin  £'  .  sin  S"  +  cos  S'  .  cos  S" 

2.  cos  S'    =  COB  A'  .  sin  S"  .  sin  S'  +  cos  S"  .  cos  S. 

3.  cos  S"  =  cos  A"  .  sin  S    .  sin  S'   +  cos  S    .  cos  S'. 

4.  cos  ^4    =  cos  S    .  sin  ./!'  .  sin  A"  —  cos  ^4'  .  cos  A". 

5.  cos  A'   =  cos  $'   .  sin  A"  .  sin  ^4    —  cos  A"  .  cos  A. 

6.  cos  A!'  =  cos  5"  .  sin  A    .  sin  A'   —  cos  J.    .  cos  A'. 

7.  cos£    .cos  ^4'   =  cot  S"  .  sin  S    -  sin  .4'   .  cot  A" 

8.  cos  £'  .  cos  A"  =  cot  S    .  sm  £'   —  sin  ^4"  .  cot  ^1 

9.  cos  S"  .  cos  A    —  cot  £'  .  sin  /S"  —  sin  ^4    .  cot  A'. 


sin  A  _  sin  A'  _  sin  A" 

sin  £  ~~  sin  Sf       sin  S" ' 


11.  sin  \  (Sr  +  5)  :  sin  }(&  —  S)  :  :  cot  £  4"  :  tan  £  (A1  -  ^4). 

12.  cos£  (£'  +  5)  :  cosi  (£'  _  ^)  :  :  cot  J  4"  :  tan  J  (^'  +  ^4). 

13.  sin  J  (^4'  +  ^4)  :  sin  \(A'  —  A)  :  :  tan  £  S"  :  tan  i  (Sr  -  ^). 

14.  cos  J  (^'  +  -4)  :  cos$  (A'  -  ^4)  :  :  tan  £  S"  :  tan  }  (£'  +  ^). 

In  these  formulae  A,  A',  A",  denote  the  several  angles  of  the  triangle ; 
and  S,  S',  5",  the  sides  opposite  those  angles  respectively.  For  the 
more  convenient  computation  of  the  formulas  Nos.  1-9,  certain  auxiliary 
angles  are  introduced,  which  will  be  alluded  to  in  the  formulae  for  the 
solution  of  the  several  cases  of  oblique-angled  spherical  triangles. 


458 


SPHERICAL   ASTRONOMY. 


VlLL.  Solutions  of  the  cases  of  right-angled  spherical  triangles. 

Required. 


Solution. 
s'n  x  ~  sin  h  •  sin 


Given. 

Hypothen 

and       <    side  adj.  giv.  ang.     2.  tan  x  —  tan  h  .  cos  a. 

an  angle.      ,,       ,, 

v.  the  other  angle.       3.  cot  x  =  cos  h  .  tan  a. 


Hypothen. 
and 


the  other  side. 


4.  cos  x  = 


cos  h 
cos  $' 


ang.  adj.  giv.  side.    5.  cos  x  =  tan  s  .  cot  h. 


sm  s 


ang.  op.  giv.  side.     6.  sin  x  =  - 

sin  h 


the  hypothen. 
the  other  side, 
the  other  angle. 

*7.  sin  x  = 
8.  sin  x  — 

sin  s 

tho  ambiguous  cases. 

sin  a 
tan  s  .  cot  a 
cos  a 

cos  * 

A  side  and 
the  angle  < 
opposite. 


A  side  and  f  tlle  tyPotlien-  10-  cot  x  =  cos  a  .  cot  s. 

the  angle  J  the  other  side.  11.  tan  x  =  tan  a  .  sin  s. 

adjacent.       ,,       ^ 

I  tne  other  angle.  12.  cos  x  =  sm  a  .  cos  s. 

The  two    (  the  hypothen.  13.  cos  x  —  rectang.  cos  of  the  given  sides. 
sides. 


an 


The  two 
angles-     1  a  side. 


the  hypothec.        15.  cos  x  =  rectang.  cot  of  the  giv.  angles 


.,  „  cos  opp. 

16.  cos  x  =  — 


sm 


ang. 


In  these  formulae,  x  denotes  the  quantity  sought. 
a  =  the  given  angle. 
s  =  the  given  si<'e. 
A  =  the  hypothenuse. 


TRIGONOMETRICAL   FORMULAE.  4.59 


IX.     Solutions  of  the   cases  of  oblique-angled   spherical 

triangles. 

GIVEN,  Two  sides  and  an  angle  opposite  one  of  them. 

Required,  1°.  The  angle  opposite  the  other  given  side. 

sin  side  op.  ang.  sought  x  sin  giv.  ang. 

sm  x  — .      . .    8 2-: f -• 

sin  side  oppos.  given  angle 

Required,  2°.  The  angle  included  between  the  given  sides. 

cot  a!  =  tan  giv.  ang.  x  cos  adj.  side, 

cos  a'  x  tan  side  adi.  giv.  ang. 

cos  a    = — : — :L-5 — : — S-, 

tan  side  op.  given  angle 

x  =  (a'  ±  a"). 

Required,  3°.  The  third  side. 

tan  a'  =  cos  giv.  ang.  x  tan  adj.  side, 

„       cos  a'  x  cos  side  op.  giv.  ang. 
cos  side  adj.  given  angle 

x  =  (a'  =b  a"). 

In  these  formulae,  x  denotes  the  quantity  sought :  af  and  a"  are 
auxiliary  angles  introduced  for  the  purpose  of  facilitating  the  compu- 
tations. 

The  angle  sought  in  formula  1  is,  in  certain  cases,  ambiguous.  In 
the  formulae  2  and  3,  when  the  angles  opposite  to  the  given  sides  are 
of  the  same  species,  we  must  take  the  upper  sign ;  on  the  contrary,  the 
lower  sign.  The  whole  of  these  formulae  therefore  are,  in  certain  cases, 
ambiguous. 

[continued. 


4:60  SPHERICAL   ASTRONOMY. 


IX.  continued.    Solutions  of  the  cases  of  oblique-angled 
spherical  triangles. 

GIVEN,  Two  angles  and  a  side  opposite  one  of  them. 

Required,  4°.  The  side  opposite  the  other  given  angle. 

sin  ang.  op.  side  sought  X  sin  giv.  side 

sin  x  = -2 — J s_ __» m 

sin  ang.  op.  given  side 

Required,  5°.  The  side  included  between  the  given  angles. 

tan  a'  =  tan  giv.  side  X  cos  ang.  adj.  giv.  side, 

sin  a'  X  tan  ang.  adj.  giv.  side 

sin  a    = =-r rt — ' — » 

tan  ang.  op.  given  side 

x    =  (a1  =fc  a"). 

Required,  6°.  The  third  angle. 

cot  a'  =  cos  given  side  X  tan  adj.  angle, 

„ sin  a'  X  cos  ang.  op.  giv.  side 

cos  ang.  adj.  given  side 

x    =  (a'  ±  a"). 

In  these  formulae,  x  denotes  the  quantity  sought:  a'  and  a"  aie 
auxiliary  angles  introduced  for  the  purpose  of  facilitating  the  compu- 
tations. 

The  side  sought  in  formula  4  is,  in  certain  cases,  ambiguous.  In 
the  formulae  5  and  6,  when  the  sides  opposite  the  given  angles  are  of 
the  same  species,  we  must  take  the  upper  sign;  on  the  contrary,  the 
lower  sign.  The  whole  of  these  formulae  therefore  are,  in  certain  cases, 
ambiguous. 

[continued. 


TRIGONOMETRICAL    FORMULAE. 


IX.  continued.     Solutions  of  the  cases  of 
spherical  triangles. 


GIVEN,  Two  sides  and  the  included  angle. 

Required,  7°.  One  of  the  other  angles. 

tan  a'  =  cos  given  angle  X  tan  given  side, 
a"  —  the  base  —  a' 

sin  a' 

tan  x    =  tan  given  an^le  x  —  -  77. 

sm  a," 

In  this  formula,  the  given  side  is  assumed  to  be  the 
side  opposite  the  angle  sought  :  the  other  known  side 
is  called  the  base, 

Required,  8°.  The  third  side. 

tan  a'  =  cos  given  angle  X  tan  given  side, 
a"  =  the  base  ~-  •  ar, 

cos  a" 

cos  x    =  cos  given  side  X  -  r  • 
cos  a' 

In  this  formula,  either  of  the  given  sides  may  be  as 
sumed  as  the  base;  and  the  other  as  the  given  side. 


In  these  formulae,  x  denotes  the  quantity  sought :  a'  and  a"  are 
auxiliary  angles  introduced  for  the  purpose  of  facilitating  the  compu- 
tations. 

If  the  side  sought  in  formula  8  be  small,  the  formula  may  not  give 
the  value  to  a  sufficient  degree  of  accuracy  *  and  some  other  mode  must 
be  adopted  for  obtaining  the  correct  value. 

[continued. 


462  SPHERICAL    ASTRONOMY. 


IX.  continued.     Solutions  of  the  cases  of 
spherical  triangles. 


GIVEN,  A  side  and  the  two  adjacent  angles. 

Required,  9°.  One  of  the  other  sides. 

cot  a'  =  tan  given  angle  X  cos  given  side, 

a"  =  the  vertical  angle  ~  a', 

cos  a' 

tan  x    =  tan  given  side  X  -  r.  . 
cos  a" 

In  this  formula,  the  angle,  opposite  the  side  sought, 
is  assumed  as  the  given  angle  :  the  other  known 
angle  is  called  the  vertical  angle. 

Required,  10°.  The  third  angle. 

cot  a'  =  tan  given  angle  X  cos  given  side, 

a"  —  the  vertical  angle  —  a', 

sin  a" 

cos  x    =  cos  mven  anme  X  -  r  . 

sm  a' 

In  this  formula,  either  of  the  given  angles  may  be 
assumed  as  the  vertical  angle  ;  and  the  other  as  the 
given  angle. 


In  these  formulae,  x  denotes  the  quantity  sought:  af  and  a"  are 
auxiliary  angles  introduced  for  tho  purpose  of  facilitating  the  compu- 
tation.*. 

If  the  angle  sought  in  formula  10  be  small,  the  formula  may  not  give 
the  value  to  a  sufficient  degree  of  accuracy;  and  some  other  mode 
must  be  adopted  for  obtaining  the  correct  value 

[continued. 


TRIGONOMETRICAL   FORMULA. 


IX.  continued.    Solutions  of  the  cases  of 
spherical  triangles. 


GIVEN,  The  three  sides. 
Required,  11°.  An  angle. 

/A  +  B  +  C         \  [A  +  B  +  C 

2  ---  B)  X  Sm  (  --  2 

sin  B.  sin  C 


sin  ,0  .  sm 


In  these  formulae,  -4,  5,  C  are  the  three  sides  of  the 
triangle  ;  and  A  is  assumed  as  the  bide  opposite  to  the 
angle  required. 


GIVEN,   The  three  angles. 

Required,  12°.  A  side. 

a  +  b  +  c 


>S 


_ 
sin  b  .  sin  c 


(a  +  b+c       T\              /a  +  b  +  c         \ 
— 6)xcos(— e) 


°\_2 

sin  6  .  sin  c 

In  these  ^tvtnulee,  a,  o,  c  are  the  three  angles  of  the 
triangle  ;  and  a  is  assumed  as  the  angle  opposite  to 
the  side  required. 

In  these  formulae,  x  denotes  the  quantity  sought.  The  formulae,  which 
are  resolved  hy  the  cosine,  are  used  only  when  the  angle  or  side  x  is 
mi  all. 


464  SPHERICAL    ASTRONOMY. 


X.  Trigonometrical  series. 


2. 


2.3.4       2.3.4.5.6 

2  a;5  17  x1 

iTT  +  F7  sTT  + 


5.  vernam  x  =  —  — 


2        2.3.42.3.4.6 

sin3 a;       1  .  3  sin5  x 


7. 


2.4.5 
cos3  x       1.3  cos5  x 


8.  a;  =  tan  x  —  J  tau?  x  +  J  tan5  ar  —  <fec. 


In  the    series  No.  7,  it   denotes    the    periphery   of    the   circle,   or 
3.14159265. 


TRIGONOMETRICAL    FORMULAE. 


465 


XI.  Multiple  arcs. 

sin  0  =  0, 

gin  x  =  sin  a?, 

«n  2  x  =  2  sin  x  .  cos  or, 

sin  3  x  =  2  sin  x  .  cos  2  a;  -f  sin  ar, 

sin  4  «  =  2  sin  a?  .  cos  3  x  -f  sin  2  a?, 


cos  0  =  1, 

cos  a;  =  cos  x, 

cos  2  #  =  2  cos  a:  .  cos  *  —  1, 
cos  3  *  =  2  cos  x  .  cos  2  a?  —  cos  *, 
ooe  4  a:  =  2  cos  x  .  cos  3  a;  —  cos  2  3 

<fec.  &c.  <fec. 


tan  a;  =  tan  a?, 

2  tan  a? 


tan  2  x  = 
tan  3  x  = 


1  —  tan*  ar' 

tan  a;  -f-  tan  2  a? 
1  —  tan  x  .  tan  2  x 


tan  a;  +  tan  8  * 
11111     *  =  1  —  tan  x  .  tan  3  at 


dec. 


Ac. 


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